The militants attacked from four sides of the city in the middle of Sunday night.
Kunduz provincial council members told Ariana news that Taliban militants have reached Zakhail and Charkh Ab area of the city.
According to the reports clashes have been erupted between the Afghan security forces and Taliban fighters.
Taliban succeed to briefly capture Kunduz province last September, but government forces backed by NATO was able to recapture the city within days.
Local officials said that roads linked to the city have been blocked by government forces.
A Taliban spokesman Zabihullah Mujahid said the group had captured several checkpoints in the city.
The attacks come ahead of a major international donor conference in Brussels on Tuesday, where Afghanistan’s international partners will discuss ongoing funding for the country.