Two U.S. servicemembers were killed in Afghanistan on Wednesday, according to a statement from the U.S. -led NATO Resolute Support mission. The next round of peace talks is scheduled to begin on June 29 in Doha.
The Taliban has refused to hold direct talks with the Afghan government, which it calls a "U.S. puppet", but has said it would talk with government officials if they attend the meetings as ordinary Afghans.
Pompeo met President Ashraf Ghani as well as General Scott Miller, who leads NATO's Afghanistan mission and the United States war effort in the country, as well as other high-level Afghan officials during the unannounced one-day visit to the Afghan capital.
Talks between Khalilzad and the Taliban have focused on USA and North Atlantic Treaty Organisation troop withdrawal and guarantees from the Taliban that Afghanistan would not again become a safe haven for terrorists to plan global attack like the airborne assaults on the United States in 2001.
In return for the withdrawal of foreign forces, the United States is demanding the Taliban guarantee that Afghanistan will not be used as a base for militant attacks.
About 20,000 foreign troops, majority American, are in Afghanistan as part of a US-led North Atlantic Treaty Organisation mission to train, assist and advise Afghan forces. They carry out near-daily attacks, mainly targeting security forces and government officials.
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"We agree that peace is our highest priority and that Afghanistan must never again serve as a platform for global terrorism".
"We've made real progress and are almost ready to conclude a draft text outlining the Taliban commitments to join fellow Afghans in ensuring that Afghan soil never again becomes a safe haven for terrorists", Pompeo said.
More than 20,000 foreign troops, a lot of them American, are in Afghanistan as part of a US-led North Atlantic Treaty Organisation mission to train, assist and advise Afghan forces.
Relations between Pakistan and Afghanistan have remained tense for years largely because of deepening trust deficit between the two neighbours.
The war there is America's longest running.
In January, the chief USA negotiator working to reach a deal with the Taliban told the New York Times that the two sides had "a draft of the framework that has to be fleshed out before it becomes an agreement".