General Mohammad Ayub Salangi, police chief of the province of Kunduz, blamed the attack on the Taliban and al-Qaeda, but he did not provide any further details.
The north is one of the quietest places in Afghanistan, and armed attacks there are rare.
Reporting from Kabul, Wali Allah Shahin, Al Jazeera's correspondent in Afghanistan, said the Taliban had claimed responsibility for the attack through Dhabih Allah Mujahid, a spokesman.
Mujahid said a military vehicle was also destroyed in the bombing.
Shahin reported quoting government sources that more than 10 Afghan soldiers were killed and six others injured.
He said Kunduz hospital sources put the number of wounded at more than 30.
The attack comes a day after a suicide bomber struck a private US security firm in the southern province of Kandahar, killing up to four Afghans working for the company and wounding another, officials said.
|Up to four Afghans were killed in a suicide|
bombing in Spin Boldak on Sunday [EPA]
The security firm, US Protection and Investigations, said the bomber was riding a motorcycle and blew himself up on Sunday near a convoy, killing two employees and wounding another.
Mohammad Asif Khan, a police officer in Kandahar's Spin Boldak district, said the attack was on a main road in the district, where the company was providing security for road construction projects.
Three guards and their driver were killed, Khan said. The differing death tolls could not immediately be reconciled.
Separately in the east, police and US-led forces attacked suspected Taliban fighters crossing from Pakistan into Afghanistan, killing 10 of them and wounding 15, the provincial governor said on Sunday.
And on Monday, the International Security Assistance Force (ISAF) said several key Taliban fighters had been killed in a series of air strikes and raids in southern Afghanistan during the past week.
The raids in the province of Helmand were part of a major Nato-led offensive launched by about 5,000 Afghan and foreign soldiers early last month.
About 37,000 Nato-led troops and a separate force of nearly 12,000 US-led multinational force soldiers are in Afghanistan to hunt down the Taliban, which is trying to topple the US-backed government in Kabul.