Khalilzad is holding informal talks with Taliban representatives in Qatari capital Doha and they have made progress, sources said.
He is seeking an agreement to reduce hostilities to get a peace deal signed that would start negotiations among Afghans on both sides of the conflict.
The Taliban have long refused to meet the Afghan government for direct negotiations, saying they would agree only to government representation in broader Afghan negotiations after the US announced the withdrawal of its troops from the country.
“The Taliban don’t have problem with person, they have problem with the establishment, an establishment which revolves around the Constitution,” Omar Daudzai, Afghan president’s peace envoy, said at an event in Kabul.
“There is no statement from Taliban saying they wouldn’t sit with Ashraf Ghani as long as he is president,” Daudzai said.
Meanwhile, Anwarulhaq Ahadi, head of New NationalFrontParty of Afghanistan, said that the negotiating team should include representatives of both the government and politicians.
He also said that there should be no precondition to negotiations.