The discussions came as the State Department ordered all family members of U.S. embassy personnel in Kyiv to leave Ukraine, citing the threat of Russian military action, and authorized some embassy employees to depart as well, according to senior State Department officials who briefed reporters on Sunday. The officials, who also spoke on the condition of anonymity because they were not authorized to comment, declined to say how many embassy personnel and family members were in the country. Thinning out staff at American embassies is a common precaution when conflicts or other crises arise that could put American diplomats in harm’s way.

In his news conference last week, Mr. Biden said he had cautioned Mr. Putin that a Russian invasion of Ukraine would prompt Washington to send more troops to the region.

“We’re going to actually increase troop presence in Poland, in Romania, et cetera, if in fact he moves,” Mr. Biden said. “They are part of NATO.”

During a phone call this month, Defense Secretary Lloyd J. Austin III warned his Russian counterpart, Sergey Shoygu, that a Russian incursion into Ukraine would most likely result in the exact troop buildup that Mr. Biden is now considering.

At the time of the phone call — Jan. 6 — the Biden administration was still trying to be more restrained in its stance on Ukraine. But after unsuccessful talks between Mr. Blinken and the Russian foreign minister, Sergey V. Lavrov, on Friday, the administration is eying a more muscular posture, including not only diplomatic options like sanctions, but military options like increasing military support to Ukrainian forces and deploying American troops to the region.

“This is clearly in response to the sudden stationing of Russian forces in Belarus, on the border, essentially, with NATO,” said Evelyn Farkas, the top Pentagon official for Russia and Ukraine during the Obama administration. “There is no way that NATO could not reply to such a sudden military move in this political context. The Kremlin needs to understand that they are only escalating the situation with all of these deployments and increasing the danger to all parties, including themselves.”

A former top Pentagon official for Europe and NATO policy, Jim Townsend, said the administration’s proposal did not go far enough.