The talks began on December 7 and were being carried out behind closed doors in the Qatari capital, reports TOLO News.
The sources said on Monday that the Taliban still considered the ceasefire agreement as a premature decision and that they have so far agreed to sign the deal only with the US.
If signed, the Taliban will agree not to attack US bases within a distance of 10 km, they added.
Meanwhile, a former Taliban commander Sayed Akbar Agha said: "The agreement is almost final. Only a few issues remain that the two sides are discussing and ceasefire is one of them."
A spokesman for Chief Executive Abdullah Abdullah's Office said that Kabul and Washington were coordinating on the peace talks.
"The reduction of violence, agreeing to a ceasefire, forming an inclusive delegation for negotiations and starting intra-Afghan negotiations are the common demands of us and our allies," he added.
The latest round of talks began after US President Donald Trump in September abruptly called them off after a Taliban attack in Kabul killed an American soldier.
US Envoy for Afghan Reconciliation, Zalmay Khalilzad has been leading the US delegation in the talks for over a year, reports Efe news.
Before flying to Qatar for the peace talks, Khalilzad held a series of meetings with Afghan leaders and politic, including President Ashraf Ghani, in Kabul.
( With inputs from IANS )