Speaking after a meeting with President Benigno S. Aquino III of the Philippines, Mr. Obama, who was in Manila for the Asia-Pacific Economic Cooperation summit meeting, said he had had “repeated discussions” with Mr. Putin about the two militaries working more closely together.

Mr. Obama’s comments are the most direct indication that the United States is prepared to work with Russia to counter the Islamic State, which is also known as ISIS, ISIL or Daesh.

The president suggested that American officials remained skeptical about whether Mr. Putin would shift his focus toward the terrorist group, even in the wake of the Paris attacks.

But he said that the Russians have been “a constructive partner” during diplomatic talks in Vienna last week, where foreign ministers agreed to a path to a cease-fire in Syria and an eventual political transition.