The timeline shared by sources came a day after Afghan President Ashraf Ghani said there has been a possible breakthrough in U.S.-Taliban talks in Qatar.
The talks had been deadlocked in part over a U.S. demand that the insurgents agree to sharply reduce violence as part of any American troop withdrawal accord.
Taliban spokesmen were not immediately available for a comment.
A western diplomat in Kabul said the U.S. negotiators were working on idea that the Taliban should agree on a reduction in violence (RIV) for at least 10 days with no major violation.
The prospective deal, he said, would call for a ceasefire period between the Afghan and U.S.coalition and the Taliban, with hopes of a peace deal being reached in the near future.
"It is after those 10 days of RIV that both sides can hold talks and firm up plans to hold intra-Afghan dialog," said the source on condition of anonymity.
There are currently about 13,000 U.S. forces as well as thousands of other NATO troops in Afghanistan, 18 years after a U.S.-led coalition invaded the country following the Sept. 11, 2001, al Qaeda attacks on the United States.
The news of a potential agreement to decrease violence comes amid continued attacks in the country.
Last month, the Special Inspector General for Afghanistan Reconstruction assessed that there had been a record-high number of attacks carried out by the Taliban and other anti-government forces during the last three months of 2019.
Senior Taliban commanders on condition of anonymity said the Taliban leaders were still adamant to observe a ceasefire without making it public, referring to the Taliban's refusal to publicly declare RIV until the U.S. signed a written agreement.
"We will stop all attacks in return for U.S. commitment to cease all their operations against us in Afghanistan," said a Taliban commander.