The assault began shortly after midnight on Saturday in several parts of the city and insurgents managed to infiltrate different areas of the region before dawn, officials said.
Police, Army commandos and special forces were deployed to vulnerable areas of the city shortly after the onslaught began, Kunduz police spokesman Sarwar Hussaini was cited as saying by Efe news.
Militants took refuge in several houses during the attack, using them as combat positions and also used civil as human shields against the security forces.
Fighting continued on Saturday in the outskirts and in some neighbourhoods of Kunduz between Afghan security forces and Taliban fighters, who were pushed back from most of the areas of the city they had captured during the night.
At least 34 Taliban fighters were killed in a counter-attack by Afghan forces, according to security officials. The Defence Ministry said Afghan security forces also suffered casualties but didn't provide exact details.
The Taliban's shadow military commander for Kunduz Mullah Wali was shot dead by a special forces sniper unit in the fighting, said the National Directorate of Security.
Rohullah Ahmadzai, spokesman for the Ministry of Defence, said that hundreds of Taliban fighters were involved in the attack. He said special forces arrived in Kunduz early on Saturday morning and the Taliban had been pushed back from several areas but was still present in at least two neighbourhoods of the city as fighters were hiding in civilian houses.
He added that "in one area Taliban fighters cowardly entered a hospital" and had taken patient hostages.
"Security forces were moving slowly but carefully in counter-attack in order not to harm civil" Ahmadzai said.
Interior Ministry spokesman Nasrat Rahimi said that 56 Taliban fighters were killed in the counter-attack and airstrikes.
"The hospital where patients were taken hostage by Taliban is now cleared and captured from the militants and several Taliban fighters killed in premises of the hospital. Patients and health workers are not harmed," he added.
The assault has kept the city shut with nearly all shops and offices closed and the streets empty of traffic. Most public services, including electricity and telecommunication, were shut down and residents in the outskirts of the city were trying to move to safe places.
The violence coincided with the ninth round of negotiations between a US delegation and the Taliban in Doha, where the two sides were trying to finalize an agreement in which the withdrawal of international forces will be a key part.
Sediq Sediqqi, spokesman for the presidential palace, said: "Taliban attacks show that unfortunately they are not accepting and interested in peace and their attack is completely contradictory to what they are saying in Doha negotiation."
( With inputs from IANS )