Earlier reports had indicated that the Taliban agreed to a short-term reduction in violence, but the sources said on Monday that the militant group has shared the new US demand with their leadership and that internal discussions were underway, TOLO News reported.
The talks between the US and the Taliban were continuing, but with pauses, according to the sources.
Echoing the sources, former Taliban member Maulana Jalaluddin Shinwari, said: "(US envoy Zalmay) Khalilzad has asked them (the Taliban) that there should be an agreement on a long-term reduction in violence.
"The Taliban are consulting about this with their leaders. There are hopes that this will have a positive result."
The new developments come as a group of Afghan politic announced a "National Reconciliation Plan", which they say will form a structure to represent the country in intra-Afghan dialogue.
Earlier this week, a group of political parties in a meeting said they were seeking a national consensus on peace, said the TOLO News report.
They called on the government to join this national consensus in order to move forward peace negotiations with the Taliban.
Earlier this month, the US and Taliban resumed the stalled negotiations, which President Donald Trump had called off in September 2019.
The two sides have been holding closed-door parleys in the Qatari capital to finalize the peace deal and gradual withdrawal of the US and NATO forces which launched a war against terror in Afghanistan 18 years ago.
While the US has been demanding a considerable drop in militant violence against the forces before signing a deal, the Taliban have been arguing that a longer and wider ceasefire with the security forces can only be part of the intra-Afghan negotiations over power-sharing at a later stage.
The US has around 11,000 troops in Afghanistan and has been gradually withdrawing forces even though the two sides have not signed a deal yet.
The Trump administration intends to draw down another 4,000 troops before they sign a peace pact with the Taliban.
( With inputs from IANS )