That seemed to agree. But as always, the devil is in the details. How long would the ceasefire last and would it begin before, after, or even during the American withdrawal? How long should the cover be sufficient for the withdrawal?
All these questions represent potential dealbreakers.
Taliban sources remained optimistic, even as Western diplomats voiced concerns that there would be no deal in this round of Doha talks. On Saturday afternoon, Sayed Akbar Agha, a former Taliban official who now lives in Kabul, maintained close contacts with the insurgents that he had just talked to them.
After nearly two decades of war, which 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 presented the withdrawal of half of the 14,000 American troops in the coming months.
Before Khalilzad, an Afghan American and former ambassador to Afghanistan, had even begun the final round of direct negotiations with the Taliban, which began on Monday and was initially scheduled to last only a few days, as was the case at previous sessions.
So far, Afghan truces have come and gone, and most observers do not believe the Afghan military can stand up to the Taliban American support. If the Americans left, who would guarantee a truce?
Ryan Crocker, a former American ambassador to Afghanistan, said it was a run on the exits.
"I can not call this a little more than an effort Put a lipstick on a US withdrawal," he said. Mr Crocker said it reminded him of the Paris Peace Talks on Vietnam.