"These six days have been the most productive of the rounds we've had with the Talibs", Khalilzad said in Qatar.
The U.S. -Taliban peace talks come as an intra-Afghan dialogue in Doha, hosted jointly by the Qatari and German governments, is scheduled for July 7-8.
Suhail Shaheen, a spokesman for the Taliban's office in Qatar, said they were also pleased with the discussions.
"We are happy with the progress and hope the rest of the work is also done. We have not faced any obstacles yet".
"It's up to the Afghans to decide when negotiations start", he said.
Afghan President Ghani, who has been conspicuously quiet about the upcoming intra-Afghan dialogue, has consistently demanded the Taliban talk directly with the government.
Afghan President Ashraf Ghani insists that no agreement to end the 18-year-long conflict in his country could be reached without his government's active involvement.
Sunday's attack comes amid peace talks between the Taliban and the Afghan government underway in Qatar.
Since then, Khalilzad has held scores of talks with the Afghan government in Kabul and overseas, with the Taliban as well as with Afghanistan's neighbors - including Pakistan, which has been accused of aiding the insurgents.
But Sunday's meeting will be the first in which a number of Afghan government officials will also be among the 64 participants traveling from Kabul.
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The Taliban have regularly interacted with Afghan opposition politicians, civil society representatives and tribal elders to discuss a future governance system in the country if it eventually strikes a peace deal with the U.S.
Negotiations have had fresh momentum in recent weeks after US Secretary of State Mike Pompeo visited Afghanistan's capital, Kabul, at the end of last month.
A vehicle bomb in central Afghanistan has killed at least eight members of the security forces and four civilians and wounded dozens, many of them students attending a nearby school.
The Taliban now control almost half of Afghanistan, and are more powerful than at any time since the October 2001 USA -led invasion.
The Taliban have steadfastly refused to talk to Ghani's government, calling it a USA puppet, but have said government officials can attend the conference as private citizens.
"The Iran deal is not the first deal we've got out of", said Khalilzad.
"Essentially the four items we have been talking about ever since we started (are) terrorism, withdrawal of foreign troops, inter-Afghan negotiations and dialogue, and ceasefire", Khalilzad told AFP.
"We can expect progress if we define progress as movement toward a troop withdrawal deal", Michael Kugelman, a South Asia expert at the Washington-based Woodrow Wilson International Center for Scholars, told DW.
This time, about 40 high-profile Afghan figures and activists will fly to Doha but will not have any official status - a condition made necessary by the Taliban's refusal to deal directly with the Western-backed government in Kabul.
About 20,000 foreign troops, a lot of them American, are in Afghanistan as part of a USA -led North Atlantic Treaty Organisation mission to train, assist and advise Afghan forces.