Taliban fighters have seized key government buildings in the northeastern Afghan city of Kunduz, leaving government forces hanging onto control of the airport and their own base, a provincial assembly lawmaker said on Sunday.

A Taliban offensive has gathered momentum in recent days, as the insurgents unleashed their forces across Afghanistan after the United States announced it would end its military mission in the country by the end of August.

On Friday the insurgents captured their first provincial capital in years when they took control of Zaranj, on the border with Iran in Afghanistan's southern Nimroz province.

And on Saturday the deputy governor of the northern province of Jowzan said the outskirts of provincial capital Sheberghan was under attack from the insurgents.

An Afghan security forces spokesman said "extremely (heavy) fighting is going on" in Kunduz, as security forces fought to defend the city, regarded as a strategic prize as it lay at the gateway to mineral-rich northern provinces and Central Asia.

But a provincial lawmaker in Kunduz told Reuters the insurgents had taken key buildings in the city of 270,000 people, raising fears that it could be the latest to fall to the Taliban.

"Heavy clashes started yesterday afternoon, all government headquarters are in control of the Taliban, only the army base and the airport is with ANDSF (Afghan security forces) from where they are resisting the Taliban," Amrudddin Wali, a member of Kunduz provincial assembly, said.

Rohullah Ahmadzai, the spokesman for the Ministry of Defence, said on Facebook that special forces were in Kunduz and had been conducting 'clearance operations' in the city to take back media offices that the Taliban had captured.

Health officials in Kunduz said that 14 bodies, including those of women and children, and more than 30 injured people have been taken to hospital.