A meeting between a senior U.S. diplomat and Taliban representatives in Doha last week to discuss a possible ceasefire ended with "very positive signals".
A former Taliban minister and ex-head of their political committee, Aga Jan Mohtism, who has maintained close contacts with the insurgent group, also confirmed a meeting in Doha between US officials and the Taliban took place earlier this week.
Taliban officials told several United States media outlets on Saturday that they have held face-to-face meetings with American diplomats in Doha last week.
'The environment was positive and the discussion was useful, ' the Taliban official said on condition of anonymity because he was not authorized to speak to the media.
The Islamist insurgency had long called for direct talks with Washington.
A senior Taliban official says the insurgent group has held the first direct talks with a US official in a preliminary discussion about future plans for a negotiated peace in Afghanistan. "These are a series of meetings for initiating formal and purposeful talks".
Reports of Monday's landmark meeting in the Qatari capital appeared in American newspapers earlier this week but neither USA officials nor the Taliban directly commented on them until now.
Ms Nauert also praised Qatar's efforts for promoting peace and stability in Afghanistan, adding that Ms Wells went there to "commend the government for their ongoing support for peace in Afghanistan".
"The only demand they made was to allow their military bases in Afghanistan", said the Taliban official.
A former Taliban minister and ex-head of their political committee, Aga Jan Mohtism, who has maintained close contacts with the insurgent group, also confirmed a meeting in Doha took place earlier this week. They also discussed Taliban participation in the Afghan government.
As recently as last week, United States officials were denying reports they were ready to speak directly to the Taliban.
He said they would first exchange prisoners and then discuss other issues that could restore peace to Afghanistan.
Another person with knowledge of the talks said the United States had pressed the Taliban side to accept the ceasefire offer for Eid-ul Adha, often known in Afghanistan as Eid-al Qurban, which this year starts on August 22. "Both sides agreed upon the continuation of the meetings and talks and another meeting is expected before Eid, but the exact time and place are not clear yet".
Any talks about a future political setup would be between the Taliban and the Afghan government, the statement said.
Earlier, a State Department official told Dawn that Washington was exploring "all avenues" to advance the Afghan peace process, and was doing so "in close consultation" with the Afghan government.
The Afghan president's office said Saturday that it welcomed any support for peace efforts.
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