Negotiations are reportedly taking place for a peaceful transition of power as people flee the city.
Taliban militants have reached the outskirts of the Afghanistan’s capital, Kabul, after taking control of most of the rest of the country.
The interior minister says negotiations have taken place to ensure a peaceful transition of power.
A Taliban statement says fighters had been ordered to remain on the edges of the capital.
The Islamist group said it did not intend to take revenge on government officials or military personnel.
The assurances come as many residents of Kabul try to reach the airport, many abandoning their cars and opting to walk because of traffic jams.
It is almost 20 years since the Taliban were ousted by a US-led military coalition.
Bagram airfield and prison, about 40km (25 miles) north of the city centre, are in Taliban hands, the militants say.
Once the largest American military facility in Afghanistan, the complex was evacuated by the US military in the dead of night on 2 July.
President Joe Biden has defended his decision to speed up the US withdrawal, saying he could not justify an “endless American presence in the middle of another country’s civil conflict”.
The US now is evacuating staff from its Kabul embassy – with people seen boarding military planes at the airport, where 5,000 US troops have been deployed to help with the operation.
Earlier on Sunday, militants took control of Jalalabad, a key eastern city, without a fight.
By Ethirajan Anbarasan, BBC World Service
Panic and fear have gripped Kabul. A colleague has told me that some government offices have been asked to evacuate and shops are closed. No explanation was given for the move.
The Taliban are poised to take Kabul – it is possible that many of their fighters could have infiltrated or are living inside the city as well.
But they are aware any fighting over Kabul will result in huge civilian casualties – and want to send a message to the international community that Taliban 2.0 are different from their earlier incarnation.
They have given safe passage to most soldiers and government officials who have surrendered in other provinces. Their psychological warfare against the security forces has worked so far – several provinces have been handed over without a fight.
They may play the same game in Kabul as Western nations evacuate their nationals. And it would not be surprising if some local Taliban leaders try to score another propaganda victory in the coming days by supervising US and UK evacuations.
The Taliban overran the city, the capital of Nangarhar province, earlier on Sunday reportedly without a shot being fired.
“There are no clashes taking place right now in Jalalabad because the governor has surrendered to the Taliban,” a local Afghan official told Reuters news agency.
“Allowing passage to the Taliban was the only way to save civilian lives.”
The news came after government’s last northern bastion, Mazar-i-Sharif, fell on Saturday.
The capture of Jalalabad means the Taliban have secured the roads connecting the country with Pakistan.
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