The Old City of Mosul is the only district still under control of the militants in the northern Iraqi city, which used to be the group’s capital in Iraq.
US-backed Iraqi troops have been battling to capture Mosul for nine months.
IS also faces defeat in its Syrian stronghold of Raqqa, nearly three years after seizing large swathes of territory in both countries.
Military commander Lieutenant General Abdul Ghani al Asadi said the storming of the Old City, was “the last chapter” in the offensive to take Mosul.
About 100,000 civilians remain trapped there with little food, water and medicine and limited access to hospitals, according to the United Nations.
IS snipers are shooting at families trying to flee as part of a tactic to keep civilians as human shields, said the UN.
Iraqi government forces regained control of eastern Mosul in January, then launched an offensive a month later on the western side that includes the Old City.
The fall of Mosul would effectively signal the end of the Iraqi part of the “caliphate” that IS leader Abu Bakr al Baghdadi declared in a speech from an historic mosque in the Old City back in 2014, covering parts of Iraq and Syria.
The Russian military recently claimed to have killed the terror chief in an airstrike in Syria last month.
However, this has been treated with scepticism given he has frequently been reported killed or wounded over the last three years.