The insurgent group said it did not plan to take the capital “by force” as it continued its rapid offensive in the country.
In just over a week, the Taliban has defeated, co-opted or sent Afghan security forces fleeing from wide swathes of Afghanistan.
The militants have taken over all major cities except for Kabul, which is the last to remain in the hands of the Afghan government.
Taliban insurgents entered the capital on Sunday and were coming in “from all sides”, according to an interior ministry official.
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.
A Taliban leader said fighters had been asked to stand at entry points in Kabul, while three Afghan officials said the group had entered the capital’s Kalakan, Qarabagh and Paghman districts.
The Islamist military group has vowed to not take control of the city “by force”, saying: “No-one’s life, property and dignity will be harmed and the lives of the citizens of Kabul will not be at risk.”
The US evacuated diplomats from its embassy in the capital as the Taliban moved into the capital.
Wisps of smoke could also be seen near the embassy’s roof as diplomats urgently destroyed sensitive documents, according to two US military officials and agency photos.
Both the US and the UK have been rushing to evacuate citizens amid the Taliban’s rapid advance in the country.
At the weekend, the insurgents captured Mazar-e-Sharif, a large and heavily-defended city that was the Afghan government’s last northern stronghold with little fighting.
The Taliban also took over the eastern city of Jalalabad, which is the capital of the Nangarhar province, without a fight. “Allowing passage to the Taliban was the only way to save civilian lives,” an Afghan official based in the city said.
The group also seized the nearby Torkham border post with Pakistan at the weekend, leaving Kabul airport the only way out of Afghanistan still in government hands.
The militant group’s rapid takeover of key parts of Afghanistan comes nearly two decades after it was forced out of power as the US invaded the country.
The US is set to remove its last troops from Afghanistan in less than three weeks, despite concerns over the Taliban’s advance.
Additional reporting by agencies
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