The militants attacked from four sides of the city in the middle of the night.
Mahmood Danish, spokesman for the provincial governor, said Afghan forces had repelled the attack.
Kunduz was briefly captured by the Taliban last September -  the first time the insurgents had taken a major city - but government forces, backed by Nato, recaptured it within days.
In southern Helmand province, militants have also taken a strategically important district to the south of Lashkar Gah, killing the local police chief, officials say.
A number of other police casualties were also reported.
Residents in Kunduz told the BBC they heard heavy fighting and helicopters flying over the strategically important city.
Mohammadullah Bahej, head of the police co-ordination office in Kunduz, said the militants had been pushed back but fighting was continuing on the outskirts of the city.
Taliban spokesman Zabihullah Mujahid said the Taliban had captured several checkpoints in the city.
"A massive operation started on Kunduz capital from four directions early this morning," Mr Mujahid said via his official Twitter account.
The capture of Kunduz by the Taliban last September was a huge blow to the country's Western-backed government. The militants abandoned the city after four days but they had proved their growing capability.
The group raided Tarin Kot, the provincial capital of Uruzgan, earlier this month.
Afghan government forces are estimated to have control over no more than two-thirds of Afghanistan.
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.
Taliban militants have launched a co-ordinated attack on the northern Afghan city of Kunduz, officials say.