Like others inside and outside government in Washington, Ms. Hill said she did not think Mr. Putin actually needed a pretext if he opted to invade because the factors driving his decision-making were long established. But the latest American military venture into Iraq may provide a talking point that would allow him to draw what analysts call a false equivalence and force the Obama administration to defend itself and argue why one intervention is legitimate and the other is not.

“He’ll always use whatever pretexts that come in handy,” said James M. Goldgeier, dean of American University’s School of International Service. “The other question is does he feel the U.S. is distracted by this other issue and it gives him the opportunity to do something in Ukraine? I just feel like he’s going to make this calculation on other grounds.”

Indeed, there were indications on Friday that Mr. Putin might be looking for a way to stand down. Nikolai Patrushev, secretary of the Russian Security Council, said Moscow wanted to de-escalate the crisis in Ukraine and mediate between the government in Kiev and pro-Russian separatists, while the Defense Ministry said military exercises on the border had been completed and some forces were returning to permanent bases, according to Russian news media reports. Still, Russian troops have pulled back from the border before without ending the crisis, and Western officials remained cautious.