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.

The firm, which operates services across the West Midlands, said it would now operate up to March 2016.
The Department for Transport (DfT) said the process of awarding a contract beyond then had not begun.
In March London Midland announced it would shed 150 jobs, while in November it said driver shortages had led to cancelled or delayed services.
Figures showed about 20% of its 1,311 daily services were delayed or cancelled throughout October.
The firm has said the 150 redundancies would focus on head office and support staff and would not affect services.
A DfT spokesman said during the extra six months it would be "working closely with the company to ensure they continue to improve the service they provide to passengers".
The process had not yet started of awarding a so-called management contract to run from March 2016 to June 2017, the department said.
It added a new franchise was planned to run from June 2017, which also has not yet been awarded.

Ganesan Nimalaruban died and several other inmates, all Tamil Tiger suspects, were reportedly injured after a failed prison mutiny in the north.
The authorities deny mistreating him, saying he died of a heart attack.
They say that the prisoners were working with Tamil separatists abroad.
Human rights lawyers working with the parents of Mr Nimalaruban have lodged a petition over the death at the Supreme Court.
They want to overturn a court ruling that he must be cremated near the capital, Colombo.
His grieving mother and father say they would prefer to give him the last rites in their home district, Vavuniya in northern Sri Lanka.
They reject assertions by the authorities that such a funeral would be a security risk.
It was in the Vavuniya prison that 32 inmates who are Tamil Tiger suspects took three prison guards hostage in late June, holding them for 19 hours in a protest against prisoner transfers.
The siege was successfully broken and all the prisoners relocated.
But rights campaigners and Tamil MPs say the prison authorities and police worked with other inmates to exact a terrible revenge, attacking them with stones and weapons, physically throwing them or breaking their legs.
Nimalaruban's parents said his body was covered in blood.
The authorities deny all the accusations.
At least one other prisoner is reported to be in a coma and in hospital.

Cameron Marshall Ward was struck by a blue Vauxhall Vectra on Court Lane in Erdington shortly after 21:10 BST on Easter Monday, 1 April, last year.
Wayne Payne was found guilty of causing death by dangerous driving.
The 31-year-old, of Somerset Road, Erdington, was sentenced at Birmingham Crown Court earlier.
Taliban militants have launched a co-ordinated attack on the northern Afghan city of Kunduz, officials say.
[NEXT_CONCEPT]
London Midland has had its franchise extended by six months, the train company has announced.
[NEXT_CONCEPT]
The parents of a prisoner who died in obscure circumstances in Sri Lanka are locked in a battle with officials who say his body cannot be taken to his home town for the last rites.
[NEXT_CONCEPT]
A driver who killed a five-year-old boy when he hit him with his car in Birmingham has been jailed for five years.