Rocket launchers, precision-guided missiles and billions of dollars’ worth of other advanced American weapons have given Ukraine a fighting chance against Russia ahead of a counteroffensive. But if even a few of the arms wind up on the black market instead of the battlefield, a Ukrainian lawmaker gloomily predicted, “we’re done.”

The lawmaker, Oleksandra Ustinova, a former anti-corruption activist who now monitors foreign arms transfers to Ukraine, does not believe there is widespread smuggling among the priciest and most sophisticated weapons donated by the United States over the last year.

“We’ve literally had people die because stuff was left behind, and they came back to get it, and were killed,” she said of Ukrainian troops’ efforts to make sure weapons were not stolen or lost.

But in Washington, against a looming government debt crisis and growing skepticism about financial support for Ukraine, an increasingly skeptical Congress is demanding tight accountability for “every weapon, every round of ammunition that we send to Ukraine,” as Representative Rob Wittman, Republican of Virginia, said last month.

By law, U.S. officials must monitor the use, transfer and security of American weapons and defense systems that are sold or otherwise given to foreign partners to make sure they are being deployed as intended. In December, for security reasons, the Biden administration largely shifted responsibility to Kyiv for monitoring the American weapons shipments at the front, despite Ukraine’s long history of corruption and arms smuggling.

Yet the sheer volume of arms delivered — including tens of thousands of shoulder-fired Javelin and Stinger missiles, portable launchers and rockets — creates a virtually insurmountable challenge to tracking each item, officials and experts caution.

All of which has heightened anxieties among Ukrainian officials responsible for ensuring weapons get to the battlefield.

“It’s impossible, honestly, to ask people to go through their stocks all the time,” said Ms. Ustinova, the chairwoman of a committee in Ukraine’s Parliament that monitors the transfer of weapons, in an interview in the streets of Warsaw last month, as she rushed to catch a train to Kyiv.

At the beginning of the war, she said, “it was just about survival, and people were just passing around Javelins” to repel a column of Russian armor that bore down on Kyiv early in the invasion. While those sorts of weapons are now routinely tracked, it’s still “very difficult” to account for small arms, like rifles, or the millions of artillery shells that the United States and its allies have sent.

The scrutiny is heightened for Javelins, Stingers and other kinds of missiles, as well as small-diameter bombs, certain types of drones, night-vision goggles and other systems being supplied to Ukraine.

But Ms. Ustinova says she has seen “zero evidence” of illicit arms transfers of the sort that would destroy Ukraine’s credibility and threaten at least a cutback in U.S. support.

“Once there is smuggling or misuse of weapons, we’re done,” she said.

So far, American officials said, there have been only a handful of cases of suspected arms trafficking or other illicit military transfers of advanced weapons sent to foreign conflicts that must be most closely tracked.

Currently, federal investigators are looking into reports of Javelin shoulder-fired rockets and Switchblade drones being sold online after being taken from Ukraine, according to an American official, who spoke on condition of anonymity to discuss a highly sensitive issue.

There was one confirmed report of a Swedish-made, anti-tank grenade launcher being smuggled out of Ukraine. But the theft was discovered only after the weapon exploded in the trunk of a car about 10 miles outside Moscow, injuring a retired Russian military officer who had just returned from eastern Ukraine.

Inspectors at the Pentagon, State Department and the U.S. Agency for International Development reported in March that they had “not yet substantiated significant waste, fraud or abuse” of American support that has been sent to Ukraine out of 189 complaints they received alleging misconduct.

A rare visit by American inspectors to a Ukrainian military facility in Odesa on April 26 found “no irregularities,” said Capt. William Speaks, a spokesman for U.S. forces in Europe.

The commander of NATO troops in Europe, Gen. Christopher G. Cavoli, told Congress late last month that he could recall only one case of attempted smuggling — some automatic rifles — since the war began. He said he remained “highly confident” in Ukraine’s ability to secure the nearly $37 billion in U.S. weapons and other security assistance that has been committed so far.

But the threat remains. In intense conflicts like the one in Ukraine, weapons are being used almost as quickly as they are received. That makes hand-held missile systems and other portable arms “vastly more difficult” to track, said Nikolai Sokov, a senior expert at the Vienna Center for Disarmament and Non-Proliferation in Austria.

Accounting for ammunition is “next to impossible,” Mr. Sokov said. He cited unconfirmed reports of Stinger missiles “roaming Ukraine free,” and said officials appeared to be trying to persuade Ukrainian citizens to return light arms they received to defend themselves last year.

“This is what happens in every large-scale, lengthy conflict, and I do not see any reason to think it may be different with Ukraine,” Mr. Sokov said.

In interviews and congressional testimony, more than a half-dozen American and Ukrainian officials described an assiduous but fallible process to track U.S.-delivered weapons.

Before they cross into Ukraine, arms shipments stop at military staging centers in Europe, where the weapons’ serial numbers are recorded into multiple databases that are viewed by American and Ukrainian officials. The serial numbers are rechecked along the delivery route into Ukraine to make sure none are missing. They are also used to identify weapons that have been lost and later reclaimed; arms that turn up far from Ukraine would indicate they were smuggled.

Ukrainian officials “track it as it goes forward,” General Cavoli told Mr. Wittman in the House hearing. “We watch over their shoulder.”

This past December, American officials began giving Ukrainian troops hand-held bar code scanners to instantly transmit the serial numbers of advanced weapons into an American database. The new process was part of the decision by the Biden administration to give Ukraine more authority to self-report how it is securing arms.

American military officials said the shift was necessary, given that fighting has largely prevented U.S. inspectors from visiting battlefield units. But American officials responsible for the oversight remain concerned they cannot personally confirm the weapons’ whereabouts.

At least some Ukrainian frontline units under constant Russian fire are still waiting to receive hand-held scanners, Ms. Ustinova said. Such battlefield assessments have been infrequent in other war zones, American officials said, as smuggling generally becomes a concern when entire containers of sensitive missiles or rocket systems go missing — not individual light weapons.

Ms. Ustinova said Ukrainian officials and troops were all too aware of the stiff criminal penalties not just for smuggling American weapons but also failing to report any losses — arms destroyed or captured on the battlefield. Each lost weapon system is investigated and its serial number reported to the U.S. Embassy in Kyiv, she said, “so in case it shows up, in Iran or somewhere, we’re not being accused of that.”

She said the 16-person committee she chairs has doggedly investigated news reports of Western arms meant for Ukraine that have supposedly turned up with gangs, terror groups and other criminals. But Ms. Ustinova said she has found no evidence those reports are true, and echoed American assertions attributing them to Russian disinformation campaigns to sow doubt about NATO support for the war.

Yet the scrutiny is wearing on Ukrainian officials, who are balancing their dire need for weapons against onerous expectations for tracking them.

President Volodymyr Zelensky of Ukraine revealed “a twinge of frustration” and an air of “How many times do I have to tell you?” when the issue was raised last month by a U.S. delegation to Kyiv, said Senator Lisa Murkowski, Republican of Alaska, who was on the trip.

But Mr. Zelensky agreed it is necessary, she said, to ensure the continued provision of American weapons and other security assistance.

“All it will take is a situation where we find that somebody, somewhere down the chain, has gotten a piece of military equipment and has sold it for personal enrichment, or misappropriated it in some way,” Ms. Murkowski said. “Because then it just gets that much harder.”

Thomas Gibbons-Neff contributed reporting.

Ukraine’s planned counteroffensive against Russia has overshadowed talk of a potential negotiated settlement in the conflict, but some U.S. and European officials say the next phase of the war could create momentum for diplomacy.

It is unclear how the officials will define success in the counteroffensive, which could last many months, or how its outcome might affect their approach. Opinions range widely among military strategists about whether Ukraine is likely to regain territory after more than a year of war.

For now, President Vladimir V. Putin of Russia has shown no signs of wanting to make concessions or engage in meaningful dialogue.

And U.S. officials remain wary of any calls for an immediate cease-fire or peace talks, especially those coming from China. Beijing persists at trying to play peacemaker, despite its obvious strategic alignment with Russia. Foreign Minister Qin Gang has been traveling across Europe this week to try to sell the notion that China can shepherd negotiations.

Some European officials meeting with Mr. Qin have expressed skepticism. And in Washington, Secretary of State Antony J. Blinken met with his counterparts from Britain and Spain this week to shore up commitments to military aid to Ukraine, sending a message that battlefield gains are the priority.

Mr. Blinken said on Tuesday at a news conference with James Cleverly, the British foreign secretary, that the Ukrainians have “what they need to continue to be successful in regaining territory that was seized by force by Russia over the last 14 months.”

Like Mr. Blinken, Mr. Cleverly did not mention diplomacy with Russia at all, instead focuseing on military aid: “We need to continue to support them, irrespective of whether this forthcoming offensive generates huge gains on the battlefield, because until this conflict is resolved and resolved properly, it is not over.”

Ukrainian leaders also say they will not agree to talks until they have pushed back Russian forces.

Still, President Biden’s aides have been exploring potential endgames, trying to identify an outcome that could be acceptable to both Kyiv and Moscow if real peace talks started, U.S. officials say.

“I know that senior-level administration officials are regularly having conversations about what peace ultimately would look like with our Ukrainian counterparts,” said Representative Adam Smith of Washington, the top Democrat on the Armed Services Committee, “while at the same time having conversations about how to arm them and win back as much territory as possible.”

Mr. Biden’s aides and European officials say their best hope is for Ukraine to make substantial gains during the counteroffensive, which would give it more leverage in any negotiations.

But whatever its leaders may think, American officials say that most Ukrainians have little appetite for compromise with their Russian attackers.

And U.S. officials fear that even if Russia’s military suffers more setbacks this summer, Mr. Putin may still believe he can win a war of attrition.

Avril D. Haines, the director of national intelligence, said in congressional testimony last week that while Mr. Putin was “scaling back his near-term ambitions” in Ukraine, the chance of Russian concessions at any negotiating table this year “will be low.”

Another senior U.S. official said that no matter what success Ukraine achieves, the Russian leader could simply order a wider draft mobilization to rebuild some of his military power.

Mr. Putin could also benefit as the 2024 presidential campaign gears up in the United States, with former President Donald J. Trump the early Republican front-runner. Mr. Trump and several Republican politicians have called U.S. support for Ukraine wasteful and dangerous.

China has pushed for a mediator role since it unveiled a vague peace initiative in February. Though Mr. Blinken and some top European diplomats say they are open to the possibility of China playing a helpful role in the future, they criticize Beijing for not publicly recognizing Russia as the aggressor in the war. They insist that a country unwilling to do that cannot be trusted to be a dispassionate mediator.

Xi Jinping, China’s leader, made a state visit to Moscow in March and voiced continued support for his nation’s partnership with Russia, which the two governments said had “no limits” just before Russia invaded Ukraine in February 2022. China’s special envoy for its peace initiative, Li Hui, was the ambassador to Russia for 10 years and received a medal from Mr. Putin.

U.S. and European officials are also suspicious of calls for peace talks that do not include a demand that the Russian military first withdraw from Ukrainian territory, which is the position of President Volodymyr Zelensky of Ukraine. China has not taken an explicit position on Ukraine’s territorial integrity, and U.S. officials say China and Russia might use the pretense of talks to freeze the front lines — and Russian gains.

In her congressional testimony, Ms. Haines said Mr. Putin could use a cease-fire to try to regain strength while “buying time for what he hopes will be an erosion of Western support for Ukraine.”

She added that “he may be willing to claim at least a temporary victory based on roughly the territory he has occupied.”

Mr. Blinken recently said it was “a positive thing” that Mr. Xi had finally spoken to Mr. Zelensky last month, but he was “still not sure” China was willing to accept that Ukraine was the victim. Annalena Baerbock, the German foreign minister, said nearly the same thing directly to Mr. Qin at a news conference on Tuesday: “Neutrality means taking the side of the aggressor, and that is why our guiding principle is to make it clear that we are on the side of the victim.”

The main argument for a greater Chinese role in diplomacy is the fact that the country is Russia’s most powerful partner, and Mr. Xi and Mr. Putin share a personal bond. Russia’s war has roiled the global economy, creating problems for China.

“As a matter of principle,” Mr. Blinken said, “countries — particularly countries with significant influence like China — if they’re willing to play a positive role in trying to bring peace, that would be a good thing.”

The White House said on Thursday that Jake Sullivan, the national security adviser, talked about Ukraine with Wang Yi, China’s top foreign policy official, during a two-day meeting this week in Vienna.

The debate in Washington over potential peace talks is amorphous and paradoxical. There are even competing arguments based on the same hypothetical outcome: If Ukraine makes substantial gains, that might mean it is time for talks, some officials say — or it could mean Ukraine should put diplomacy on the back burner and keep fighting.

If Ukraine is unable to seize significant territory, some U.S. and European officials might want to nudge Mr. Zelensky toward a negotiated settlement.

“The dynamic will shift even if Ukraine makes marginal gains,” said Mr. Smith, the Democratic lawmaker. After several more months of war, he predicted, both sides will be exhausted.

But some officials and analysts in Washington caution against such thinking.

“There’s always been a desire among some people in Washington to say, look, if Ukraine doesn’t make gains — or if they do — it might be time to have a conversation about Ukraine looking for a settlement,” said Alina Polyakova, the president of the Center for European Policy Analysis.

“I personally find that shocking,” she added. “Territorial concessions would validate Russia’s aggression, which sets a global precedent for China and others that such means work. Two, it would also mean that the West would have to accept the moral implications — accepting war crimes and condoning continued human rights abuses.”

Among top U.S. officials, Gen. Mark A. Milley, the chairman of the Joint Chiefs of Staff, has been the most outspoken on the need for Ukraine and Russia to consider negotiations, arguing that a prolonged war would result in many more casualties. Mr. Blinken has taken a different position. “There has to be some profound change in Mr. Putin’s mind and in Russia’s mind to engage in meaningful diplomacy,” he said last week.

The secretary of state and other American officials have made vague statements on what they see as a viable end to the conflict.

At least twice in the past several months, Mr. Blinken has referred to the need for Ukraine to reclaim territory “seized by force by Russia over the last 14 months,” as he put it on Tuesday. But years before this invasion, Russia seized effective control of hundreds of square miles of eastern Ukraine and annexed its Crimean Peninsula in March 2014.

It is unclear whether Mr. Blinken is intentionally drawing a distinction between those swaths of territory. Ukrainian leaders insist their goal is to reclaim every inch of their land taken since 2014, including Crimea. But many U.S. officials and analysts believe that Mr. Putin would take more drastic measures to retain his grip on the peninsula.

Some U.S. officials have raised the possibility of at least forcing Russia to demilitarize Crimea, so that it cannot be used as a staging ground for future attacks on Ukraine. But that outcome could be almost as difficult for Mr. Putin to accept. The Russian Black Sea Fleet is based at the Crimean city of Sevastopol.

Mr. Blinken said last week that a “just and durable” peace plan “can’t ratify what Russia has done, which is the seizure of so much of Ukraine’s territory.” Nor can it allow Russia to “simply rest, refit and reattack six months later or a year later.”

Julian E. Barnes contributed reporting from Washington, and Steven Erlanger from Brussels.