In particular, Khalilzad told the New York Times that the two sides had reached an agreement in principle on a framework for a peace deal that would see foreign troops withdraw from Afghanistan over a period of 18 months.
Following six days of talks between U.S. and Taliban delegations in Doha, Qatar, the framework agreement involves the insurgent group agreeing to prevent Afghan territory from being used by terrorists such as the Islamic State (IS) group.
American and Taliban officials agreed on the framework of a deal that includes US forces leaving the area in exchange for a pledge from the Taliban that their territory in Afghanistan would not be used by them or other terrorist groups to harm the USA or its interests, The New York Times reported.
Khalilzad's statement emphasized the inclusion of the Afghan government in the talks.
More than 45,000 Afghan security personnel have "paid the ultimate sacrifice" since 2014, Afghan President Ashraf Ghani said during the World Economic Forum's Annual Meeting in Davos, Switzerland, last week.
"We need to get a stable Afghanistan as an entity that can ensure security of America and Europe and others on the one hand, but more fundamentally our own democratic rights and institutions and our right to live in peace and harmony", Ghani told CNN's Fareed Zakaria.
"We have a strong will that we'll bring about peace to end Afghanistan's 40 years of war and to develop and rebuild the nation", Ghani said.
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The statement from Ghani's office also claimed that the Taliban demanded from Khalilzad the withdrawal of all foreign troops from Afghanistan, but that there was also no agreement on that issue. It comes as the Taliban continues to stage near-daily attacks against the Western-backed Afghan government. To the contrary, this is the same Taliban that enabled the September 11 attacks by giving al Qaeda a safe haven in Afghanistan from which to operate; that triggered the American-led invasion on October 7, 2001, after refusing to hand over Osama bin Laden; and that continues a close alliance with al Qaeda to this day. "No Afghan wants to face suicide attacks in hospitals, schools, the mosques, and parks", said Ghani.
Khalilzad said that the talks in Qatar have been "more productive than they have been in the past", referencing years of without any progress between the US and the Taliban.
The Taliban refuse to meet directly with the Afghan government, which they view as a USA puppet. No Afghan wants foreign troops to remain in their country indefinitely.
The Taliban have insisted foreign troops must pull out. Many Americans will applaud this "progress" as one step closer to bringing the 14,000 American troops now deployed in Afghanistan home.
"We have to make it clear to ourselves", Glaser says, "that a ground presence in Afghanistan doesn't protect us from terrorism".
US Special Representative for Afghanistan Reconciliation Zalmay Khalilzad, who met with Afghan media in Kabul on Monday, said there is a "moment of opportunity for peace" and that he hopes Afghans will deal with the opportunity "urgently and positively".
"There is a lot more work to be done before we can say we have succeeded in our efforts but I believe for the first time I can say that we have made significant progress", he said.