The timing is notable. If the seven-day period begins this weekend or early next week, and it holds for a week, the peace agreement would be ready to sign around the time that President Trump is scheduled to travel to India. That opens the possibility that the president, who previously attempted to bring members of the Taliban to Camp David for a signing, would have the option of traveling to a secure location, like Bagram Air Base in Afghanistan, to formalize an agreement.

A truce has been widely expected, and Mr. Pompeo and Mr. Esper met on Friday in Munich with Ashraf Ghani, Afghanistan’s president. But there was no announcement of an imminent deal, and similar efforts have fallen apart. Mr. Trump pulled back on an accord last year after an attack that killed an American soldier and 11 others.

The senior official who briefed reporters in Munich said that the agreement would not go into effect until a verified, successful reduction in hostilities, which is far short of a blanket cease-fire.

The violence reduction deal, the official said, is highly specific, saying that it would have to be nationwide and include violence against Afghans as well as members of the American-led coalition, and would apply to all the elements of violence that have become familiar in 18 years of war: roadside bombs, suicide bombs and rocket attacks.

The pace of an American pullout is the issue of most political importance to Mr. Trump. One of the few issues on which he and the Democratic challengers for president agree is the need for the United States to get out of Afghanistan — though many of the candidates said in a survey published by The New York Times last week that they would keep a small counterterrorism force and some intelligence presence.