These talks mark the highest level negotiations between the two sides since the US ramped up peace efforts last year.
Cofounder of the Taliban, Mullah Abdul Ghani Baradar, has joined the talks as the new head of the Taliban team. He met the US special representative Zalmay Khalilzad for the first time on Monday.
Since then, the two men and their respective delegations have been working to hammer out details of a framework agreement reached in six days of negotiations in Doha in January.
“We have two core issues. The withdrawal of all foreign forces from Afghanistan is a core issue for us. And a core issue for the American side is that the soil of Afghanistan should not be used against the Americans and against its allies,” said Taliban spokesman Suhail Shaheen as talks entered a second day on Tuesday.
The US has also been asking for a Taliban ceasefire.
“If we do not reach a solution in this round of talks, then we will in the next round of talks, but that is our target,” Shaheen said.
This round of talks, now in their fourth day, after a two-day recess, have working groups trying to find formulas that work for both sides.
One major sticking point is the US wants the Taliban to negotiate any final deal with the Afghan government led by President Ashraf Ghani, something the Taliban has repeatedly refused to do, calling the government a “puppet” of the West.
Ghani has expressed frustration at being excluded from the negotiations. Taliban members met Afghan representatives, including Ghani’s political opponents in Moscow last month.
The Taliban continue their military offensives in Afghanistan, despite the talks.
Taliban fighters killed at least 23 Afghan security forces in an attack on Friday on Camp Shorab, a sprawling military base in Helmand province, southern Afghanistan.
In northern Sari Pul province, Taliban ambushed a convoy of Afghan security forces, killing nine, said Zabihullah Amani, spokesman for the provincial governor.
The heavy fighting has been costly for Afghan security forces. While the Afghan Ministry of Defence stopped publicly releasing casualty figures, Ghani said 45,000 Afghan security forces have been killed since September 2014.
Last year was the deadliest for civilians in Afghanistan according to the United Nations with 3,804 civilian deaths in 2018.
The UN says at least 32,000 civilians have been killed and another 60,000 wounded in the past decade, when the organisation began compiling the data.