Obama and his closest foreign policy advisers laid out the reasons for his military commanders. Keeping as many as 10,000 troops in Afghanistan indefinitely at a cost of as much as $10 billion to $15 billion a year wasn’t politically feasible or financially responsible. There were more pressing domestic priorities that needed money. The country faced bigger threats.

Then, in August, Gen. Martin E. Dempsey, the outgoing chairman of the Joint Chiefs of Staff, came in with one more plan to maintain a counterterrorism force of as many as 5,000 troops in Afghanistan to prevent a reemergence of al-Qaeda and to battle Islamic State fighters seeking a foothold in the country. Dempsey’s plan was a quick, ­back-of-the-envelope exercise, according to senior administration officials.

This time, though, Obama didn’t dismiss it. “I think that’s an argument that can be made to the American people,” Obama said, according to a senior administration official who took part in the meeting and who, like others interviewed for this article, spoke on the condition of anonymity to discuss internal deliberations.