The US and the Taliban thrashed out elements of a deal to bring a close to Afghanistan's 18-year conflict at the second day of renewed talks in Doha on Sunday.
The US, which invaded Afghanistan and toppled the Taliban in 2001, wants to withdraw thousands of troops and turn the page on its longest war.
But it would first seek assurances from the insurgents that they will renounce Al Qaeda and stop other militants such as ISIS using the country as a haven.
The talks, now in their eighth round, began on Saturday with no end date given publicly.
A Taliban source said efforts had been made to organise a direct meeting between US envoy for Afghanistan Zalmay Khalilzad and Taliban co-founder Mullah Baradar, who leads the movement's political wing.
The men have met before, as recently as May.
A coalition led by Washington ousted the Taliban in late 2001, accusing it of harbouring Al Qaeda militants who claimed the September 11 attacks against the US, which killed almost 3,000 people.
But despite a rapid conclusion to the conventional phase of the war, the Taliban have proved formidable insurgents, bogging down US troops for years.
Washington is hoping to strike a peace deal with the Taliban by September 1, before Afghan polls due the same month and US presidential elections due in 2020.
"We've made a lot of progress. We're talking," US President Donald Trump said on Friday.
Mr Khalilzad tweeted on Friday: "We are pursuing a peace agreement, not a withdrawal agreement – a peace agreement that enables withdrawal.
"Our presence in Afghanistan is conditions-based and any withdrawal will be conditions-based."
He was commenting as he arrived in Doha after talks with Pakistani Prime Minister Imran Khan in Islamabad.
In another sign of progress, the Afghan government has formed a negotiating team for separate peace talks with the Taliban that diplomats hope could be held as early as this month.
An initial deal to end the war would see the US force in Afghanistan reduced to as low as 8,000 from the current level of about 14,000, The Washington Post reported on Friday.
In exchange, the Taliban would abide by a ceasefire, renounce Al-Qaeda and talk to the Kabul administration.
"After 19 years, President Trump has made it very clear that his desire is that we develop a diplomatic resolution that permits us to reduce the resources that are located there, while simultaneously ensuring that Afghanistan never again becomes a platform where a terrorist can strike the United States," US Secretary of State Mike Pompeo said on Sunday during a visit to Sydney.
An Afghan official hinted last week that the government of President Ashraf Ghani was preparing for direct talks with the Taliban, the details of which have not been announced.
"We have no preconditions to begin talks but the peace agreement is not without conditions," Mr Ghani wrote on his Facebook page on Friday, before the talks.
"We want a republic government not an emirate," he said, in a challenge to the Taliban.
"The negotiations will be tough and the Taliban should know that no Afghan is inferior in religion or courage to them."
The thorny issues of power-sharing with the Taliban, the role of regional powers including Pakistan and India, and the fate of Mr Ghani's administration also remain unresolved.
The latest US-Taliban encounter follows last month's talks between influential Afghans and the Taliban, which agreed on a "road map for peace" but stopped short of calling for a ceasefire.
Kabul resident Somaya Mustafa, 20, said her country desperately needed a peace deal but only one in which the Taliban "accept women and their achievements".
"It is a total mess in our country right now," Ms Mustafa said. "And if it continues, women will suffer more than anyone else."
The UN has said that civilian casualty rates across Afghanistan matched record levels last month, after a dip earlier in the year.
