The Taliban are proposing a three-month ceasefire in return for the release of about 7,000 insurgent prisoners held in Afghanistan's jails.
According to a member of the Afghan negotiating team who spoke to The National, discussions are pending.
"They've offered a ceasefire, but that will mean Afghan forces can't retake lost districts," the negotiator said. "But, most importantly, we have had an extremely sad experience of releasing prisoners in the past. The violence following the last prisoner release was extreme."
Nader Naderi, who is also a member of the Afghan government negotiating team, said earlier on Thursday that the insurgents want the names of the movement's leaders to be removed from a United Nations blacklist.
The offer comes as the Taliban tighten their grip on the country, seizing towns along major routes and border crossings vital for trade. Residents of Spin Boldak in Kandahar province said Taliban fighters entered the town on Wednesday.
The crossing between Spin Boldak and Chaman is the landlocked country's second busiest entry point and main commercial artery between its south-west region and Pakistan's sea ports. Afghan government data shows that the route is used by 900 lorries a day.
The Afghan government said it was still in control of the area, but Pakistani officials in Chaman reported seeing the Taliban flag flying from the Friendship Gate at the border.
Afghan interior ministry spokesman Tareq Arian said the Taliban's attempt to take Spin Boldak had failed.
"The terrorist Taliban had some movements near the border area ... the security forces have repelled the attack," he told AFP.
The seizure of the area follows the Taliban's capture last week of the Islam Qala crossing on the border with Iran, the country's busiest trade point.
They have in recent days seized other major border crossings in Herat, Farah and Kunduz provinces in the north and west.
Control of border posts allows the Taliban to collect revenue, said Shafiqullah Attai, chairman of the Afghanistan Chamber of Commerce and Investment in the capital Kabul.
"Income has started to go to the Taliban," Mr Attai told Reuters, but he could not estimate how much they were earning.
The Taliban, who ruled Afghanistan from 1996 until they were ousted in 2001 by a US-led invasion following the September 11 attacks, have been fighting since then to topple the western-backed government in Kabul.
Emboldened by the departure of foreign forces, to be completed by a September deadline, the militants are making a new push to surround cities and capture territory while continuing to hold peace talks with the government.
The Taliban agreed to negotiate with the government as part of an agreement under which the US offered to withdraw its forces. The deal also involved the release of 5,000 Taliban prisoners by the government.
Meanwhile, in the US, former president George W Bush criticised President Joe Biden's decision to withdraw from Afghanistan.
“I'm afraid Afghan women and girls are going to suffer unspeakable harm … they are going to be left behind to be slaughtered by these very brutal people and it breaks my heart,” Mr Bush told German broadcaster Deutsche Welle.
Asked whether he thought the withdrawal was a mistake, Mr Bush replied: “Yes, I think it is.”
Mr Bush was speaking before German Chancellor Angela Merkel's visit to Washington, where she will meet Mr Biden on Thursday.
Germany has maintained a troop presence in Afghanistan since shortly after the US invaded the country in 2001, and Mr Bush told Deutsche Welle he believes Ms Merkel “feels the same way” about the US withdrawal.
The pace of the US pull-out leaves many Afghans worried that hard-won social gains, such as the right for girls to go to school, will be quickly crushed by the Taliban, who are rapidly snatching territory across Afghanistan.
The hardliners were known for their brutal repression of women when they ruled much of Afghanistan during the late 1990s until the US-led invasion.
About 350 Afghans fled into Tajikistan on Wednesday from northern Afghanistan to escape Taliban attacks, the Tajik news agency Khova reported. It said the refugees, a majority of them girls, had “fled from the Taliban to save their lives”. The agency said that that two babies died during the border crossing.
Meanwhile, Washington said thousands of interpreters and others who helped US and Nato forces in Afghanistan will be removed, beginning in late July.
In what the White House is calling Operation Allies Refuge, the interpreters and their families are likely to be taken first to American overseas military bases before resettlement in the US or elsewhere.
Many want to leave the country because of fears of retaliation by the Taliban, who are seeking to regain control of the government in Kabul after the departure of US troops.
Some estimates say there are about18,000 people who would qualify to be relocated and who, with their families, could take the total of those offered refuge as high as 100,000.