“I will say that we’re skeptical at this time of the veracity of these reports,” said Matthew Miller, a State Department spokesman. He reiterated that the United States did not “encourage or enable strikes inside of Russia, and we’ve made that clear.”

“But as we’ve also said,” he added, “it is up to Ukraine to decide how to conduct this war.”

The use of U.S. military equipment on Russian soil could strain relations between Ukraine and the United States, which has given Ukraine tens of billions of dollars in military aid, with one of the conditions being that it not be used to attack Russia within its own borders.

Photos and videos showed dozens of pro-Ukraine fighters using the vehicles in their convoy as they crossed the Russian border into the village of Kozinka on Monday morning. The Ukrainian military’s involvement in the operation is not clear.

While there have been attacks over this border during the 15 monthlong war, Monday’s assault was unique in its brazenness and duration. The two units that claimed responsibility for the incursion are the Free Russia Legion and the Russian Volunteer Corps, which are made up of Russian nationals who have been fighting in Ukraine against the Russian military.

Russia has cast members of the assault force as Ukrainian militants and saboteurs; Ukraine’s deputy defense minister, Hanna Maliar, called them “Russian patriots” participating in an “internal Russian crisis.”

The Free Russia Legion is part of a unit overseen by Ukrainian officers, but Mykhailo Podolyak, a presidential adviser, said Ukraine had nothing to do with the incursion.

The assault into Russia has stretched into its second day. On Tuesday afternoon, Russia’s Ministry of Defense said it had pushed the fighters back over the border, but people who claim to represent the groups maintained that they were continuing attacks in Russia.

The incursion came days after a Friday announcement that President Biden agreed to allow Ukrainian troops to be trained on F-16 warplanes and was open to other countries supplying them to Ukraine, a reversal after a year of denying Ukrainian requests for the aircraft over concerns it could be used to strike targets within Russia and potentially escalate the conflict.

Christoph Koettl, Dmitriy Khavin and Julian Barnes contributed reporting.