The fall of the city would, in effect, mark the end of the Iraqi half of the "caliphate" declared nearly three years ago by Islamic State (IS) leader Abu Bakr al-Baghdadi, which also covers parts of Syria.
The enclave includes the Old City center and three adjacent districts along the western bank of the Tigris River.
The U.S.-backed offensive in Mosul , now in its eighth month, has taken longer than planned as the militants are dug in among civilians.
An Iraqi military statement said, "The joint forces have begun liberating the remaining districts."
Another military statement announced the death of two Iraqi colonels during the fighting on Saturday.
Desperate civilians trapped behind Islamic State lines now face a harrowing situation with little food and water, no electricity and limited access to hospitals.
The Iraqi air force dropped leaflets on Friday urging residents to flee but humanitarian groups say they fear for the safety of those trying to escape.