According to reports, shortly after launching the raid in the early hours of Monday morning the Taliban had taken control of several roads in the city, and surrounded some police stations in the city.
Taliban spokesmen also claimed to have control of a hospital and some government buildings.
There were conflicting reports about the number of casualties. The Tolo news agency reported that 21 people had been killed, and 47 injured, but that fierce fighting prevented many more people from receiving medical attention.
The Pajhwok news agency reported that at least four residents of the city were killed as a result of artillery and gunfire clashes.
A spokesman for Kunduz police expressed confidence that government could repel the attack and said that 20 Taliban fighters were killed and three police officers wounded.
"Right now heavy fighting is ongoing in Khanabad, Chardara and at Imam Saheb, the main entrances to the city," stated Sayed Sarwar Hussaini. "We have enough forces and will drive them out soon."
The Taliban last attempted a full-scale assault on Kunduz in April, following the withdrawal of most foreign troops. The strategically important city is the capital of Kunduz province, through which major roads pass connecting central and northern Afghanistan, and a main road to the capital, Kabul.