That much seemed in agreement. But as always, the devil is in the details. How long would the cease-fire be, and would it start before, after or even during the American withdrawal? How long a time frame would the withdrawal cover?

All of these questions are potential deal-breakers.

Taliban sources remained optimistic, even as Western diplomats expressed concern that there would not be a deal from this round of Doha talks. On Saturday afternoon, Sayed Akbar Agha, a former Taliban official who now lives in Kabul but keeps close contacts with insurgent leaders, said he had just spoken to them.

“I have been told that talks are going ahead very well and we are close to an agreement,” he said. Whatever the result this time, Mr. Agha said, “Afghanistan was never so close to peace in these past years. What is happening now has never happened before.”

After nearly two decades of a war that has cost tens of thousands of lives and hundreds of billions of dollars, President Trump has made clear his distaste for a continued role in Afghanistan. In December, he detailed the withdrawal of half of America’s 14,000 troops in the coming months.

That was before Mr. Khalilzad, an Afghan-American and former ambassador to Afghanistan, had even begun the latest round of direct negotiations with the Taliban, which started on Monday and were initially expected to last only a couple days, as previous sessions had.

Afghan truces up to now have come and gone, and most observers do not believe that the Afghan military can stand against the Taliban without American support. Once the Americans leave, who would guarantee any truce?

Ryan Crocker, a former American ambassador to Afghanistan, said it was a rush for the exits.

“I can’t see this as anything more than an effort to put lipstick on what will be a U.S. withdrawal,” he said. Mr. Crocker said it reminded him of the Paris peace talks on Vietnam.

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