During the first year of Russia’s war in Ukraine, the Biden administration fretted constantly that if Kyiv hit back inside Russian borders, President Vladimir V. Putin would retaliate against not only Ukraine, but also possibly NATO and the West.

But those fears have ebbed. As Ukraine’s counteroffensive edges closer, a series of bold attacks in Russia, from a swarm of drone attacks in Moscow to the shelling of towns in the Belgorod region bordering Ukraine and an incursion into the country using American-made armored vehicles, have been greeted by the Biden administration with the diplomatic equivalent of a shrug.

“It’s not like we’re going to go out and investigate this,” John F. Kirby, a National Security Council spokesman, said last week, in a reference to whether Ukraine or Ukrainian-backed groups were behind the attacks in Moscow. On Monday, fighters attacked at least 10 villages in the Belgorod region with heavy shelling, its governor said.

Behind closed doors, senior administration officials have seemed even less fazed. “Look, it’s a war,” one senior Pentagon official said last Thursday. “This is what happens in a war.”

American officials view the cross-border attacks as preliminary operations for Ukraine’s possibly unfolding counteroffensive, a sign that it will have multiple phases. The operations, they say, are an important test of Russian defenses and a flexing of muscles ahead of the big military push.

That is a far cry from the administration’s tiptoeing last year, when American officials took pains to make sure they were not giving Ukraine weaponry that could hit inside Russia, citing escalation fears. “We are not encouraging or enabling Ukraine to strike beyond its borders,” President Biden said last May in a guest essay in The New York Times, just two months after he scuttled a European proposal to send MIG-29 fighter jets to Ukraine. “We’re not going to send to Ukraine rocket systems that strike into Russia.”

Fast-forward 12 months, and Mr. Biden has signed off on sending Ukraine F-16s, an equally lethal fighter jet.

So what happened?

Since the early days of the invasion, Russia’s battered military has shown itself unable to make significant gains against Ukraine, and a wider conflict would risk drawing the United States and NATO even more deeply into the war. And fears that Russia might use a tactical nuclear weapon appear to have receded somewhat, although officials warn that could change if Mr. Putin feels cornered.

“I think the administration has really turned the corner to understanding that not only is Russia the strategic loser, but that they are very likely going to be the military loser,” said Evelyn Farkas, the top Russia and Ukraine Pentagon official during the Obama administration and the executive director of the McCain Institute.

Dr. Farkas said that the fears of escalation remain, but that “while they are real, they are not as frightening as Russia somehow prevailing.”

American military officials say the reality of warfighting is that it makes no sense to constantly play defense and fight an enemy on one’s territory alone, without putting a foe’s own home at risk.

“If you’re in a war, you can’t just sit back and give the initiative to the enemy,” said Frederick B. Hodges, a retired lieutenant general and the former commanding general for U.S. Army forces in Europe. “Under the U.N. charter, every nation has the right to defend itself, so for Ukraine, from a legal standpoint and from a military standpoint, it makes great sense.”

Officially, Biden administration officials continue to say that they do not want Ukraine to use American-supplied weaponry to carry out attacks inside Russia, either by Ukrainian troops or paramilitary groups.

“We don’t encourage, we don’t enable and we don’t support strikes or attacks inside Russia,” Mr. Kirby said on Monday at the White House. “Our effort is to support them in their self-defense, in defending their territory, their sovereignty.”

U.S. officials say that while the threat of nuclear escalation is not gone, Ukraine’s cross-border operations are not the type of action that is likely to provoke the use of a nuclear device. American intelligence officials have said they believe Russia would use a tactical nuclear device only if Mr. Putin’s hold on power was threatened, its military began to completely collapse in Ukraine or it faced the loss of Crimea, which Russian forces seized in 2014.

But concerns remain that a miscalculation or mistake by pro-Ukrainian operations could transform a symbolic attack inside Russia into something more damaging, something that the Kremlin would feel it needed to respond to more strongly or that could generate tensions and disagreements among European allies opposed to any effort by Ukraine to expand the war, according to U.S. officials.

U.S. officials also now say it is unlikely that Ukrainian attacks in Russia would prompt a Russian strike on a NATO country or facility. Mr. Putin wants to make sure the war does not spill over into other countries, which could prompt even greater U.S. involvement or spur the Biden administration to send armaments to the Ukrainians that it has been reluctant to give, for fear that they would use them inside Russia, the officials said.

Of course, Mr. Biden has begun doing so anyway, from providing Ukraine with M1 Abrams tanks to the F-16s.

Several current and former senior American, European and Ukrainian officials said the recent cross-border incursions by pro-Ukrainian forces into Russia and drone strikes around Moscow marked the beginning of Kyiv’s long-planned counteroffensive.

These preliminary attacks — what military analysts call shaping operations — are intended to disrupt Moscow’s battle plans, pull Russian troops away from the main battlefields and undermine the Russian citizenry’s confidence in the country’s forces, the officials said in interviews. They spoke on the condition of anonymity to discuss the planned offensive.

The attacks have escalated in recent weeks after strikes in Crimea and other parts of occupied Ukraine against Russian railways, supply lines, fuel depots and ammunition stores.

Michael Kofman, the director of Russian studies at CNA, a research institute in Arlington, Va., said the cross-border operations had two main objectives. “The first is to bring the war to Russia and show that it is not invulnerable,” he said. “The second is to get Russian forces to take seriously the problem of defending their border, and to get them to commit resources, perhaps pulling in troops from elsewhere.”

Mr. Kofman added, “These types of operations are low cost relative to their strategic impact and effectively magnified by Ukrainian information operations.”

One of the last things Mr. Putin wants is to have the Russian public worried that the war could come to its doorstep, two officials said.

But the Biden administration is walking a fine line. While administration officials urge Ukraine not to use U.S.-provided weapons to strike Russia on its own soil, they have also said it was up to President Volodymyr Zelensky of Ukraine and his military commanders to decide how they will use that equipment.

“We don’t tell them where to strike. We don’t tell them where not to strike,” Mr. Kirby told reporters last week. “We don’t tell them how to conduct their operations. We give them equipment. We give them training. We give them advice and counsel. Heck, we even do tabletop exercises with them to help them plan out what they’re going to do.”

Britain, another major Ukrainian ally, went further.

Its foreign minister, James Cleverly, said last week that Ukraine had “the right to project force beyond its borders” to undermine Russian attacks and that military targets beyond a nation’s borders were “internationally recognized as being legitimate as part of a nation’s self-defense.” Mr. Cleverly said he did not have details about the drone attacks and was speaking more generally.

Military analysts played down the possibility that the increasingly brazen and frequent strikes inside Russia could escalate the Kremlin’s response.

Last year’s escalation fears, General Hodges said, were “way, way overstated” by the administration, especially worries that Russia would retaliate against the West or NATO. But he noted that Russia had retaliated against Ukrainians.

“As time has moved on, with Russia continuing to kill innocent Ukrainians, with precision weapons against apartment buildings, our continued tapping of the brakes on this made us look naïve,” General Hodges said.

U.S. officials say that for now Russia has responded, sometimes forcefully, to the cross-border attacks but has not escalated the war or unleashed any sort of new response to the operations.

American officials say they believe Russia will not escalate as long as the Ukrainian strikes continue to be mostly symbolic and do not destroy critical infrastructure or targets of national importance.

The one target that the Ukrainians hit last year was of national importance and a piece of critical infrastructure: the Kerch Strait Bridge connecting Crimea to the mainland. Russia responded to that attack by beginning a campaign against Ukraine’s power grid, a notable escalation in the war.

But other than the bridge, the strikes that the United States believes were carried out by Ukraine or Ukrainian-aligned groups in Russian border cities or were targeting supporters of the Russian government have had more symbolic impact than direct impact on the war.