The US and the Taliban were thrashing 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 ever war.

But it would first seek assurances from the insurgents that they will renounce Al Qaeda and stop other militants like the Islamic State group using the country as a haven.

The talks, now in their eighth round, began on Saturday with no end date issued publicly.

A Taliban source told AFP efforts had been made to organize a direct meeting between US envoy for Afghanistan Zalmay Khalilzad and Taliban co-founder Mullah Baradar, who heads the movement's political wing.

The men have met previously, as recently as May.

A coalition led by Washington ousted the Taliban in late 2001 accusing it of harboring Al Qaeda jihadists who claimed the September 11 attacks against the US that 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 - ahead of Afghan polls due the same month, and US presidential elections due in 2020.

US President Donald Trump told reporters at the White House on Friday that "we've made a lot of progress. We're talking."

"We are pursuing a peace agreement not a withdrawal agreement, a peace agreement that enables withdrawal," Khalilzad tweeted on Friday as he arrived in Doha after talks with Pakistani Prime Minister Imran Khan in Islamabad.

"Our presence in Afghanistan is conditions-based, and any withdrawal will be conditions-based."

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 later this month.