US military planes were supporting the Afghan operation, striking Taliban positions on the outskirts of the city, a Nato spokesman said.
A Pentagon spokesman said the US had faith the Afghan military would recapture the city.
The Afghan health ministry said Kunduz hospitals had received 16 bodies, with almost 200 people injured.
Medical charity Medecins Sans Frontieres said its hospital in the city had been overwhelmed, with scores of patients suffering gunshot wounds.