Heavy fighting between Afghan security forces and Taliban militants on Monday afternoon continued following relative calm, with parts of the city falling to the Taliban, local officials claimed.
"Police district 2 and 3 have fallen to the Taliban while fighting continues near the provincial governor's palace and police headquarters inside the city," Aminullah Ayaddin, a provincial council member, said.
The violence comes just days before a conference on Afghanistan in Brussels, where world leaders are expected to pledge economic assistance for the impoverished country's development.
Afghan forces were responding from the ground and the air to the Taliban attack in Kunduz, said Adam Khan Mateen, a brigade commander for the Afghan army in the province.
The Taliban launched their attack on the north-eastern city of Kunduz from several directions early Monday. Police spokesman Mahfozullah Akbari said that special forces have since been able to push back the militants from several areas outside the city.
"US enablers, including air power, are in position and prepared to assist in Kunduz as needed," said NATO's Resolute Support mission in a tweet.
At least one policemen was killed and four others have been wounded in the fighting, the Afghan Interior Ministry said.
In September 2015, the Taliban took the strategically important city for two weeks, which came as a shock to the Afghan government and to the international community.
It was the first time the insurgents had taken over a major urban centre since the US-led offensive overthrew the Taliban regime 15 years ago.