Northern Alliance troops have entered Kunduz and say Taliban fighters are surrendering "continuously".
No fighting was initially reported as the Alliance moved in.
The top Taliban commander is among those who have surrendered, the Alliance claims.
It is hopeful the takeover of the besieged city will be complete by nightfall.
Spokesman Ashraf Nadeem, speaking by satellite telephone from near the city's western front, says Alliance commander Mir Alam entered Kunduz at midday.
Later about 2,500 troops of another alliance commander, Gen Rashid Dostum, reportedly took control of the city centre after entering from the west.
The report by the Afghan Islamic Press could not be independently confirmed.
It says Gen Dostum himself is expected to arrive in the city later in the day.