Taliban fighters were reaching the capital "from all sides", a senior Afghan interior ministry official told Reuters. However, there were no reports of fighting.
The group was in talks with the Western-backed government for a peaceful surrender, Taliban spokesman Zabihullah Mujahid said. "Taliban fighters are to be on standby on all entrances of Kabul until a peaceful and satisfactory transfer of power is agreed," he said in a statement.
Ali Ahmad Jalali, a US-based academic and former Afghan Interior Minister, could be named head of an interim administration in Kabul, three diplomatic sources said, though it was unclear whether the Taliban had agreed.
Known during their past rule for keeping girls out of school and their hard-line practice of Islamic law, including punishments of amputation, stoning and hanging, the Taliban appear to be trying to project a more modern face.
Another spokesman Suhail Shaheen said the group would protect the rights of women, as well as freedoms for media workers and diplomats.
"We assure the people, particularly in the city of Kabul, that their properties, their lives are safe," Shaheen told the BBC, saying a transfer of power was expected in days.
The ease of the Taliban's advance, despite billions of dollars spent by the United States and others to build up local Afghan government forces, has stunned the world.
Just last week, a US intelligence estimate said Kabul could hold out for at least three months.
There was no immediate word on the situation from President Ashraf Ghani. A palace official said he was in emergency talks with US peace envoy Zalmay Khalilzad and officials from the NATO transatlantic alliance.
Power would be handed over to a transitional administration, the government's acting interior minister, Abdul Sattar Mirzakawal, tweeted on the Tolo news channel. "There won't be an attack on the city, it is agreed that there will be a peaceful handover," he said without elaborating.
A tweet from the Afghan presidential palace account said firing had been heard at a number of points around Kabul but that security forces, in coordination with international partners, had control of the city.
Many of Kabul's streets were choked by cars and people either trying to rush home or reach the airport, residents said.
"Some people have left their keys in the car and have started walking to the airport," one resident told Reuters by phone. Another said: "People are all going home in fear of fighting."
Afghans had fled the provinces to enter Kabul in recent days, fearing a return to hard-line Islamist rule.
Early on Sunday, refugees from Taliban-controlled provinces were seen unloading belongings from taxis and families stood outside embassy gates, while the city's downtown was packed with people stocking up on supplies.
US officials said diplomats were being ferried by helicopters to the airport from its embassy in the fortified Wazir Akbar Khan district. More American troops were being sent to help in the evacuations after the Taliban's surge brought the Islamist group to Kabul in a matter of days.
"Core" US team members were working from the airport, a US official said, while a NATO official said several European Union staff had moved to a safer location in the capital.
Earlier on Sunday, Taliban captured the eastern city of Jalalabad without a fight, giving them control of one of the main highways into landlocked Afghanistan. They also took over the nearby Torkham border post with Pakistan, leaving Kabul airport the only way out of Afghanistan still in government hands.
"The governor has surrendered to Taliban," a Jalalabad-based Afghan official told Reuters. "Allowing passage to Taliban was the only way to save civilian lives."
After US-led forces withdrew the bulk of their remaining troops in the last month, Taliban campaign accelerated as the Afghan military's defences appeared to collapse.
President Joe Biden on Saturday authorized the deployment of 5,000 US troops to help evacuate citizens and ensure an "orderly and safe" drawdown of military personnel. A US official said that included 1,000 newly approved troops from the 82nd Airborne Division.
Taliban said its rapid gains showed it was popularly accepted by the Afghan people.
