An official declaration of victory from the government is expected soon.
Iraqi forces, backed by US-led air strikes, have tried to retake the city since 17 October last year.
IS seized Mosul in June 2014 before sweeping across much of Iraq's Sunni Arab heartland and proclaiming a "caliphate" straddling Iraq and Syria.
But they have been losing ground over the past nine months, as government forces advance on their former Iraqi stronghold.
Kurdish Peshmerga fighters, Sunni Arab tribesmen and Shia militiamen, supported by US-led coalition warplanes and military advisers, have been involved in the battle.
On Saturday the jihadists were desperately holding out in a tiny area near the Old City. State television said troops had expected to take full control within hours.
Jonathan Beale, BBC defence correspondent, Mosul
After months of intense fighting that's gutted the city, the sound of gunfire is more sporadic. Coalition warplanes are still flying overhead, but there's a lull in the bombardment.
The Iraqi security forces have still been facing pockets of resistance, but they're already claiming victory against Islamic State.
That's not yet been confirmed by the Iraqi government or the US-led coalition - but an announcement is expected soon.
It'll be a significant moment - it's taken nearly nine months to oust the extremists from the city that was once their stronghold.
But even their defeat here will not mean the end of IS in Iraq.
The government announced the full "liberation" of eastern Mosul in January, but the west of the city has presented a more difficult challenge, with its narrow, winding streets.
Last October, the Iraqi army said there were 6,000 militants in the city. Fewer than 300 were thought to be holding out.
Some 900,000 people have been displaced from the city since 2014 - about half the the pre-war population- aid organisations say.
Last month, Iraqi Prime Minister Haider al-Abadi said the destruction of the ancient mosque in the city of Mosul was "an official declaration of defeat" by IS.
Iraqi forces say IS blew up the Great Mosque of al-Nuri and its famous leaning minaret as jihadists battled to stop advancing pro-government troops.
The Iraqi army has been mopping up the last pockets of resistance from Islamic State (IS) militants in Mosul, after a long battle to recapture the city.