Talks to end America’s longest and most expensive war began again at the weekend between US and Taliban negotiators.
The meeting is the ninth round about the future of America's role in Afghanistan and the possible end to hostilities between the US and the insurgent Taliban, 18 years after the invasion toppled the hardline group’s administration.
"Talks resumed this afternoon," a US source told AFP on Saturday.
The two sides were set to meet on Friday but "both decided to postpone it until Saturday", Taliban spokesman Suhail Shaheen said.
"We have made progress and are now discussing the implementation mechanism and some technical points. The agreement will be completed after we agree on these points."
Any deal would be announced before the media, representatives from neighbouring countries, China, Russia and the UN, Mr Shaheen said.
The US wants to withdraw thousands of troops but only in return for the insurgent group renouncing Al Qaeda and curbing attacks.
Zalmay Khalilzad, the US special envoy to Afghanistan, has led the talks and says he is ready to conclude peace talks with the Taliban.
Mr Khalilzad said last week that he hoped this would be the last year of the Afghan war.
Washington is hoping to strike an agreement with the Taliban by September 1, before Afghan polls due the same month, and US presidential polls due in 2020.
US President Donald Trump vowed to bring troops home in his campaign for election in 2016.
The Taliban claimed on Saturday to have killed seven members of the US military in an attack on a convoy near Bagram airfield north of Kabul.
Col Sonny Leggett, a spokesman for US forces in Afghanistan, said some local businesses had been damaged in the attack but denied coalition or US troops were hurt.
"Rather than taking responsibility for, and apologising for, the harm they bring to innocent Afghans and their businesses, the Taliban spread lies about the attack," Col Leggett said.
On Wednesday, two US soldiers were killed by small arms fire in Faryab province in northern Afghanistan, the Pentagon said.
Mr Shaheen said the deaths should have a "positive" impact on the talks in Doha.
The deaths last Wednesday bring the total number of US soldiers killed in Afghanistan this year to 17, up from 13 last year and 11 in 2017.
As of last year, 2,372 US military personnel had been killed in the war.