"The military council of the Taliban has agreed on a reduction of violence. They are waiting for the signature of the (group's) Emir," former member Mawlawi Abdul Shakoor Mutamen told TOLO News on Monday.
The second member, former commander Sayed Akbar Agha, also told the news portal that the "Taliban has agreed to consider a reduction of violence in response to the call for a ceasefire. They will agree on this... and then they (the US and the Taliban) will discuss the peace deal".
But the Afghan government has said that its absence from the US-Taliban talks in Doha, Qatar makes a settlement difficult.
"We hope that the talks (in Qatar) lead to direct discussions between the Afghan government and the Taliban," acting Foreign Minister Idris Zaman said.
"Ceasefire is one of the conditions for a first step and we are still calling for it."
Meanwhile, the Afghan High Peace Council has demanded an "explanation" by US chief negotiator Zalmay Khalilzad for the delay in the next round of the talks, which were halted after the Taliban attacked a health facility near the Bagram Airfield in the north of Kabul last month, which left eight people dead including six attackers and injured over 70 civil, the TOLO News report said.
"How come the Taliban are insisting on not finalizing the talks, and why are the Americans not willing to sign it? This should be explained by Khalilzad," a member of the council said.
Marathon talks initiated between the US and the Taliban in October 2018 in Doha to find a negotiated settlement for Afghanistan's lingering crisis, broke down in September following a Taliban-linked car bomb that killed 10 people including an American soldier in Kabul.
The talks resumed on December 7, 2019 but was again suspended in the wake of the Bagram attack.
( With inputs from IANS )