At the top of that list of difficult questions is certain to be the status of next month’s planned meeting between Mr. Obama and Mr. Putin, which has been in doubt for weeks because of issues that include Russia’s refusal to return Edward J. Snowden to the United States to face charges of leaking national security secrets.

American officials have signaled that Russia’s decision to grant Mr. Snowden temporary asylum undermined the relationship between the countries and could affect whether Mr. Obama agrees to travel to Moscow next month. He is scheduled to be in St. Petersburg for a meeting of the Group of 20 scheduled for Sept. 5 and 6, and had planned to go from there to meet Mr. Putin.

On Monday, Jay Carney, the White House press secretary, said a decision was forthcoming.

“I think it’s fair to say that you can expect that we’ll have a decision to announce in coming days about that specific issue,” Mr. Carney told reporters. “We obviously disagree with the Russians very strongly about the decision they’ve made on Mr. Snowden.”

But Mr. Carney said the administration also disagreed with Russia on “a number of other issues, including Syria.” And he said that the question of whether the two presidents met would be determined by whether such a gathering could help make progress on a broader set of concerns.

“We are evaluating that against not just our disagreement over Mr. Snowden, but some of the other issues where we have failed to see, thus far, eye to eye,” Mr. Carney said. “And once we have fully assessed the utility of a summit, we’ll make an announcement.”