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 told the BBC's Mahfouz Zubaide 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.
The Taliban said they had also entered a hospital in Kunduz, although local media reports say they only stayed briefly.
Pictures on social media appeared to show fighters taking selfies inside the hospital.
Kunduz province contains major roads that connect central and northern Afghanistan. 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 said it carried out that attack.
Also on Sunday, 300 fighters allied to the so-called Islamic State attacked checkpoints in Nangarhar province.
Two policemen were reported killed in the assault, while local officials said 60 militants were killed.
Hundreds of Taliban fighters have stormed the strategic northern Afghan city of Kunduz, seizing control of half of it, police say.