It is not clear how Russia will react to the fortified military presence along NATO’s eastern flank. Since the signing of a cease-fire agreement last year, Mr. Putin’s government has tried to ease tensions with the West. Officials said the Russian government was eager for the United States and Europe to roll back economic sanctions, which suggested that it would not escalate tensions over the new military commitments.

But outside analysts were surprised by the magnitude of the increase in military funding for Europe, which is part of an overall budget request of $580 billion for the Pentagon. Mr. Obama, according to a defense official, is also going to ask Congress for a 35 percent increase — $7 billion — to fight Islamic State militants.

Some analysts said the increased funding and deployments would certainly rattle Russia. Among the countries where the equipment and additional forces could be deployed are Hungary, Romania and the Baltic countries, Pentagon officials said.

“This is a really big deal, and the Russians are going to have a cow,” said Evelyn N. Farkas, who until October was the Pentagon’s top policy official on Russia and Ukraine. “It’s a huge sign of commitment to deterring Russia, and to strengthening our alliance and our partnership with countries like Ukraine, Moldova and Georgia.”

While the increase in funding for Europe is significant, the administration is proposing that the money come from a separate war-funding account that is meant to pay for operations in the conflicts in Iraq and Syria, as well as the continued American military presence in Afghanistan. That means it is a one-time request, not necessarily a continuing commitment built in to budget requests beyond 2017, officials said.