Sher Mohammad Abbas Stanikzai, who until recently was the head of the Taliban's political office in Qatar, spoke to the BBC while attending a meeting in Moscow with a number of senior Afghan opposition politicians.
"This meeting was very successful", the Taliban official told reporters, flanked by the former president who was appointed after US-led forces routed the Taliban in 2001.
The considerable obstacles that remain include, most obviously, the Taliban's refusal to speak to the Afghan president, Ashraf Ghani, and his government. Ghani said that Pompeo also "underscored the central importance of ensuring the centrality of the Afghan government in the peace process" and signaled support for holding Afghan presidential elections in July.
Khalilzad is in a hurry to find a peace deal for Afghanistan that would allow America to bring home its troops after 17 years of war. Now the exact timetable for the pullout must be decided, but the Taliban wants to see this happen in months, he said.
The Taliban routinely exaggerates its attacks and details of other engagements, and North Atlantic Treaty Organisation and Afghan officials deny the claims.
The U.S. promised to withdraw half of its troops by April, the Russian state-run RIA Novosti news agency reported Wednesday, citing a Taliban representative in Moscow.
He said that "when the American forces announce the withdrawal of their troops" there could be further "intra-Afghan dialogue".
In a tweet, Khalilzad addressed "claims" made by Taliban officials that the U.S. had a troop pull-out timetable for Afghanistan.More news: Instagram's record breaking egg uses fame for good
"A group of Taliban rebels stormed the checkpoints of Local Police in Baghlan-e-Markazi district early this morning triggering a gun battle, which left 10 local police personnel killed", the official told Xinhua but declined to be named, saying authorized officials would talk to the media.
"The people who are in Moscow, including the Taliban and Afghan politicians, have one thing in common: they are all against Ghani", Yunus Fakur, a Kabul-based Afghan analyst, told DW. "Those taking part in the talks are independent individuals", Ghani said, according to Tolo News. Taliban meet USA officials and attend a peace conference in Moscow.
This comes after a senior Taliban member and deputy head of their Qatar office said on Monday, after the first day's session, that the group would talk to the Afghan government once a timeline was set for the complete withdrawal of USA forces from Afghanistan.
He also mentioned U.S. President Donald Trump's remarks on his State of the Union address on Tuesday, in which he said the U.S. withdrawal from Afghanistan was tied to the development of the peace process.
"We need to make sure everything they say here, they mean it", Ms Koofi added.
A Taliban official said on Wednesday that no timetable had been agreed with the US government for the partial withdrawal of USA forces from Afghanistan, and that negotiations were still underway.
Meanwhile, as the talks were underway in Moscow, violence continued in Afghanistan. "Let hundreds of such meetings be held, but these would only be paper (agreements) unless there is an agreement by the Afghan government; Afghanistan's national assembly and Afghanistan's legal institutions", he stressed.
The Taliban have been staging near-daily attacks, inflicting heavy casualties on the embattled Afghan army and security forces.