"These six days have been the most productive of the rounds we've had with the Talibs... we made progress on all the issues that we have been discussing," Khalilzad said in Qatar.
The insurgents have been meeting with the US envoy in Doha to try to forge a deal that would see the US military quit Afghanistan in return for various guarantees.
Suhail Shaheen, the spokesman for the Taliban's office in Qatar, said they were pleased with the discussions so far.
"We are happy with progress and hope the rest of the work is also done. We have not faced any obstacles yet," he tweeted.
But the talks are set to be paused for two days while another summit with Afghan representatives takes place in Qatar's capital.
'They want peace'
"Tomorrow and the day after, because of the intra-Afghan conference, no talks will take place," Shaheen said.
US envoy Khalilzad confirmed direct US-Taliban talks will restart on Tuesday.
About 60 Afghan delegates are expected to attend Sunday's so-called intra-Afghan dialogue, including political figures, women and various Afghan stakeholders.
Meanwhile, People in Afghanistan follow the developments anxiously.
“People in Kabul and other cities see it positively,” Abdullah Kahn, Managing Director of the Pakistan Institute for Conflict and Security Studies in Islamabad told RFI.
“They are fed up with the fighting that’s been going on for a decade. They want peace in Afghanistan.
Regional militant groups
But, he points at the “complex” situation in the country with many regional militant groups, such as the Islamic Movement of Uzbekistan which was formed in 1998 by Islamic ideologue Tahir Yuldashev and former Soviet Paratrooper Juma Namangami, and affiliated with Al Qaeda, the Turkestan Islamic Party, an Uyghur group that has its headquarters in North Waziristan, but which operates in Afghanistan as well, which destabilize the region.
“The problem is: will the Taliban be able to control these groups? This will be a big challenge to them,” a topic that will constitute a “serious discussion” in the upcoming dialogue between the US and the Taliban in Doha.
The big question is if the Taliban wants to dissociate itself from Al Qaeda, responsible for the 2001 9/11 twin tower attack in New York. “How the Taliban will technically dissociate itself from Al Qaeda is a major problem,” he says.
Meanwhile, the Taliban, who have steadfastly refused to negotiate with the government of President Ashraf Ghani, have stressed that those attending the talks planned for Sunday and Monday will only do so in a "personal capacity".
But the US has made it clear the Taliban must talk with the Afghan government before a peace deal can be completed.
US Secretary of State Mike Pompeo has said he hopes for an agreement before September 1.