An eyewitness told the BBC that he had seen the Taliban advance. Fierce fighting has been going on all day.
The defence ministry has denied the city is in militant hands.
Kunduz was briefly captured by the Taliban in September 2015 - the first time the insurgents had taken a major city - but government forces, backed by Nato, recaptured it within days.
The Taliban have published a video apparently confirming they are back inside the city.