President Ashraf Ghani was reported to have fled the country as the Taliban drew closer to victory.
Officials earlier said talks had taken place to ensure a peaceful transition.
Thousands of Afghans have sought refuge in Kabul in recent weeks. There were scenes of panic in the city on Sunday as people tried to flee.
Western countries have also been scrambling to evacuate their citizens. The US sent military helicopters to transport staff from the heavily fortified embassy compound in Kabul to the airport.
A Taliban spokesman told the BBC there would be “no revenge” on Afghans.
It is almost 20 years since the Islamist group was ousted by a US-led military coalition. They have been seizing territory across the country at speed now that foreign troops have all but withdrawn.
US President Joe Biden has defended the US withdrawal, saying he could not justify an “endless American presence in the middle of another country’s civil conflict
There is panic in Kabul, where some residents have been trying to reach the airport to leave the country. Cars have been abandoned and people have opted to walk because of traffic jams.
Residents have also been rushing to withdraw cash from ATMs, and queuing to get travel documents at the passport office and at foreign visa centres.
Farzana Kocha, an MP in Kabul, told the BBC that people did not know what to do as Taliban militants closed in on the city.
“Some of them are running, some are hiding in houses,” she said. captionKabul MP: “Some people are running, some are hiding in houses”
The Taliban said its fighters had been ordered to remain on the edges of the capital for now.
Earlier on Sunday, they said they had taken control of Bagram airfield and prison, about 40km (25 miles) north of the city centre.
Once the largest American military facility in Afghanistan, the complex was evacuated by the US military in the dead of night on 2 July.
Taliban spokesman Suhail Shaheen told the BBC that people in Kabul had no need to worry and that their properties and lives were safe.
“We are the servants of the people and of this country,” he said.
He added that the group did not want Afghans to flee, but instead to stay and help with the post-conflict reconstruction.