"Meetings here (in Qatar) were more productive than they have been in the past. We made significant progress on vital issues", Zalmay Khalilzad, the USA special representative for Afghan reconciliation, wrote on Twitter. This makes an agreement between the Taliban and the USA more likely now than at any other time. Taliban spokesman Zabihullah Mujahid confirmed to AFP that "discussions are still ongoing".
The next step, the sources acknowledged, will be face-to-face talks between the Afghan government and the Taliban.
The Taliban said that "the withdrawal of foreign troops from Afghanistan and other vital issues saw progress".
The sources also said that Taliban has agreed to not let al Qaida and Daesh operate in Afghanistan.
He said that USA, as a friendly country, should fulfill its promise to quit Afghanistan and added that it should not be tug of war for power in Kabul after the withdrawal of U.S. forces.
In December, reports emerged that the United States was planning to withdraw about 7,000 troops - roughly half the remaining USA military presence in the country.
"Nothing is agreed until everything is agreed, and "everything" must include an intra-Afghan dialogue and comprehensive ceasefire".
Officials at the USA embassy in Kabul were not immediately available to comment on the diplomatic developments.
Qatari officials have denied reports that Afghan intelligence chief Mohammad Massoum Stanikzai flew to Doha to participate in the talks.
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Based on unconfirmed reports, Taliban's representatives will visit Kabul soon to talk with the Afghan government.
Baradar, formerly a number two of the group, helped Mullah Omar, who died in 2013, to found the Taliban movement in Afghanistan in 1994.
The Taliban regime, which ruled most of the country from 1996 to 2001, brutally enforced its religious code against women working or studying, and curtailed rights and liberties for all Afghans.
Abdul Ghani Baradar, a senior Taliban military commander, was arrested in Pakistan in 2010. "We wish that the USA leaves Afghanistan as a friend of the region, not as a failure", Pakistan army spokesman Major-General Asif Ghafoor said before the Doha talks". He later joined his family in Afghanistan.
However, according to the sources, Taliban has not responded to this call by Khalilzad.
"[He] is considered as someone who can build bridges with the different factions within the Taliban, in particular, with the Shura council of the Taliban, the entity which will have the final say over any political future of Afghanistan", Ahelbarra said. In return, the Taliban have said they will not allow Afghan territory to be used for attacks on the USA or other countries.
They were driven from power by US-led troops following the 9/11 attacks which Washington blamed on al-Qaeda militants sheltered by the Taliban.
Yao Jing, China's ambassador in Islamabad, also praised Pakistan's role in facilitating the Afghan peace process - including its efforts to facilitate talks between the Taliban and United States - and helping end the long conflict in Afghanistan.
It further relayed that representatives of Pakistan and hosts Qatar also attended the dialogue.