Afghan President Ashraf Ghani has left the country bound for Tajikistan, reports say.
A Taliban spokesman says their fighters have been ordered to enter the city to prevent looting, reports BBC.
Earlier they were told to refrain from violence and allow safe passage for anyone wanting to leave.
Peaceful transfer of power to transitional government being prepared, acting interior minister says.
Eyewitnesses say the militants met little resistance along the way to the capital.
The Taliban capture more territory, including the former US airbase at Bagram and the central Bamiyan province
UK PM Boris Johnson is to recall Parliament from its summer break on Wednesday to discuss the crisis.
US has started evacuating staff from its embassy in Kabul.