Taliban fighters mounted a coordinated assault on the northern city of Kunduz overnight, attacking from four directions and entering the city itself, a senior city police official said.
Sheer Ali Kamal, commander of the 808 Tandar police zone in Kunduz, said the attack began at around midnight (1930 GMT Sunday) and fighting was still going on in and around the city.
"We are putting all our efforts together to push them back," he said.
Military helicopters were flying overhead and gunfire could be heard in the city.
Kunduz, which fell briefly to the Taliban a year ago, has seen repeated bouts of heavy fighting as the insurgents have sought to repeat their success of 2015.
Monday's attack, a day before the start of a major donor conference in Brussels, underlines the precarious security situation in Afghanistan, where government forces are estimated to have control over no more than two-thirds of the country.
"A massive operation started on Kunduz capital from four directions early this morning," Taliban spokesman Zabihullah Mujahid said in his official Twitter account.
He said the Nawabad area with four checkpoints had been captured and a number of soldiers had been killed. It was not immediately possible to verify the claim.
A Reuters reporter saw at least five Taliban fighters armed with AK-47 assault rifles, machines guns and rocket-propelled grenades in the city.
The attack came as the Taliban have stepped up operations in different parts of Afghanistan, including the strategic southern province of Helmand, where they have been threatening the provincial capital of Lashkar Gah.
The fall of Kunduz last year was one of the most serious blows suffered by the Western-backed government in Kabul since the withdrawal of most international troops at the end of 2014.
Although the insurgents abandoned the city after a few days, the demonstration that they were able to take a provincial capital underlined their growing strength and exposed serious flaws in Afghan security forces.
A Taliban raid on Tarin Kot, the provincial capital of Uruzgan in the south, on Sept. 8 also sparked fears of another collapse like that in Kunduz last year.
Mujahid said in the Taliban statement that the militants were pressing ahead with their assaults on Helmand and Uruzgan.
Afghanistan's international partners are due to start a two-day conference in Brussels on Tuesday, where they are expected to approve maintaining billions of dollars in funding for the government over the next four years.