Militants have occupied some government buildings, including a prison, and heavy fighting is continuing.
One report said the Taliban had raised their flag in the city centre.
The government said at least 25 militants and two Afghan policemen had been killed and that reinforcements had been sent to the city.
Monday's attack appears to be one of the most significant mounted on a provincial capital by the Taliban, correspondents say.
Kunduz police spokesman Sayed Sarwar Husaini said that militants had captured the jail in Kunduz and freed about 500 prisoners, including members of the Taliban.
Kunduz province has seen a number of attacks since April, with the Taliban joining forces with other insurgents.
Reports said the Taliban attacked the city from three sides at about dawn. A witness told Reuters news agency that by mid-afternoon they had raised their white flag in the city's main square, about 200m (218 yards) from the governor's compound.
Battles were raging in two districts close to the compound, the witness said.
"The Taliban have taken over our neighbourhood... I can see their fighters all around," an AFP news agency journalist inside the city reported.
The city is symbolically significant to the Taliban as it was their former northern stronghold before their government was overthrown in 2001.
The assault comes a day after a bomb attack on a sports match in eastern Paktika province killed nine people.
