Taliban assault Afghanistan's Kunduz city from three directions
The exact number of casualties is not clear, but media reports suggest that several people have been killed.
It is the second time this year the militants have attempted to breach the provincial capital.
It comes a day after a bomb attack on a sports match in the eastern Paktika province killed nine people.
Kunduz police spokesman Sayed Sarwar Hussaini told journalists that the Taliban attacked the city from several directions in the early hours of Monday.
"Heavy fighting is ongoing in Khanabad, Chardara and at Imam Sahed, the main entrances to the city," he told Reuters. "We have enough forces and will drive them out soon."
There were conflicting reports on the number of casualties.
Local officials told Tolo News agency that 21 people died in clashes with the Taliban, while 13 Taliban fighters were killed.
Meanwhile, police told AP that 20 militants had died.
Kunduz province contains major roads that connect central and northern Afghanistan, including a road to the capital Kabul.
Correspondents say it also supplies half of Afghanistan's rice crop.
Militant violence has increased across Afghanistan since the departure of most US and Nato forces last year.
On Sunday, a bomb attack on a sports ground in Paktika left nine dead and dozens injured. No group has claimed responsibility for the attack.
Also on Sunday, 300 fighters allied to the so-called Islamic State attacked checkpoints in Nangarhar.
Two policemen were reported killed in the assault, while local officials said 60 militants were killed, Kazinform refers to BBC.com.
