The draft deal was agreed after six days of talks in Qatar last week.
The Islamic State group is also a potent, growing force in Afghanistan, where it is fighting a fierce turf war with the Taliban and has claimed responsibility for many devastating attacks, especially in Kabul.
The US invasion of 2001 was driven by the Taliban's harbouring of Al Qaeda, but more than 17 years later the militant group appears diminished in the region.
USA and Taliban officials have agreed in principle to the "framework" of a peace deal, The New York Times quotes US special envoy Zalmay Khalilzad as saying after five days of talks between the militant group and the United States in Qatar.
Khalilzad will stay in a Kabul for more talks with the Afghan government Monday, and there are discussions that Khalilzad may go back to Afghanistan in early February.
The Taliban demanded from Khalilzad the withdrawal of all foreign troops from Afghanistan, but that there was no agreement on that, the statement added.
Khalilzad did not confirm the statement - or his remarks to the Times - and there was no immediate comment from the U.S. Embassy in Kabul.
After six days of talks in Doha, Qatar, Khalilzad few to Kabul to consult with the Afghan government.
Acting US Defence Secretary Patrick Shanahan said he was encouraged by US peace talks with Afghan Taliban insurgents but added he had not yet been tasked with planning a full withdrawal of troops. "With the withdrawal of foreign forces from Afghanistan, a civil war will erupt", said Wahid, a Facebook user.
Ghani also reiterated his call on the Taliban to engage "in direct talks with the Afghan government".
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Afghan president Ashraf Ghani called on the Taliban to "enter serious talks" with the government in Kabul Monday, following unprecedented marathon negotiations between the insurgents and the USA in Qatar last week.
"We want peace quickly, we want it soon, but we want it with prudence", Ghani said.
"No Afghans want foreign forces in their country for the long term", Ghani said.
However, there is still no accord on a timetable for a United States withdrawal or a ceasefire - major issues on which previous attempts at negotiations have foundered.
Trump is planning to withdraw more than 5,000 of the 14,000 USA troops in Afghanistan, a US official said in December, in the latest sign that the US president's patience is thinning with America's longest war and overseas military interventions.
He reassured the Afghans the talks in Qatar remain geared towards bringing the insurgents to the table with Kabul, according to a statement released by Ghani's office.
According to Mr Ghani's office, Mr Khalilzad in a meeting on Sunday denied that there had been any discussions with the Taliban about future governance arrangements in Kabul. Trump last month rebuffed top advisers and made a decision to pull US troops out of Syria as well.
"The Taliban have committed, to our satisfaction, to do what is necessary that would prevent Afghanistan from ever becoming a platform for global terrorist groups or individuals", he told the Times.
Australian combat troops were withdrawn in December 2013, but 400 personnel remain in Afghanistan as trainers and advisers.
"Pakistan played a vital role in bringing the USA and the Taliban to the negotiations table".
Afghan security forces are already taking staggering losses, with 45,000 killed since late 2014, and morale is low.