Mohammad Yusouf Ayubi, the head of the provincial council in Kunduz, said the group's forces attacked the city from multiple directions in the small hours on Saturday.
Hundreds of people fled their homes as the sound of gunfire and explosions echoed from the city's outskirts, he was quoted as saying by the Associated Press news agency.
Naeem Mangal, the head of the Kunduz regional hospital, told the AFP news agency that at least eight people were killed and 62 wounded.
Also on Saturday, the Taliban claimed responsibility for an attack on a district headquarters in the eastern Nangarhar province late on Friday.
At least two Afghan troops were killed in the incident, along with 27 Taliban fighters, according to Attahullah Khogyani, the governor's spokesman.
Nangarhar Governor Shahmahmood Miakhel said Afghan forces repelled the attack after reinforcements arrived.
The Taliban, which frequently exaggerates numbers, claimed to have killed or wounded "more than 200 soldiers, police and militias".
The assault came hours after the Taliban announced the start of Operation Fath, the name the group has given to this year's spring offensive.
Also on Friday, the Taliban ambushed a police convoy in the western Ghor province, killing seven security forces, according to Abdul Hai Khateby, the provincial governor's spokesman.
Among those killed was Faqir Ahmad Noori, the head of operations for the provincial police. Another two police and a civilian were wounded, Khateby said.
In the northern Baghlan province, the Taliban killed seven police and wounded eight in an attack on checkpoints late on Friday, said Safder Mohsini, the head of the provincial council.
Afghan President Ashraf Ghani's office on Saturday condemned "in the strongest words" the announcement of Operation Fath.
"The continuation of war is no one's interest," it said.