US-backed Iraqi forces pushed deeper into the last pocket of Mosul controlled by Islamic State militants on Sunday as the battle for the city approaches an end after seven months of gruelling urban combat.

The militants have now been dislodged from all but a handful of districts in the western half of Mosul including the Old City, where Islamic State is expected to make its last stand, taking advantage of narrow streets and its dense population.

Brigadier General Yahya Rasoul said the area controlled by Islamic State was no more than 9 percent of west Mosul, which is bisected by the River Tigris.

'It's a very small area,' he told Reuters. 'God willing, this is the final phase.'

The elite Counter Terrorism Service (CTS) stormed the Ureibi and Rifaie districts at dawn on Sunday, according to a statement from the Joint Operations Command.

At the same time, the army's ninth division and the Interior Ministry's elite Emergency Response Division attacked the Islamic State bastion of 17 Tammouz.

'Daesh (Islamic State) is drawing its last dying breath,' the commander of the ninth division, Lieutenant General Qasim Nazzal, told state television on Sunday. 'Daesh fighters are broken and quickly retreating from fronts.'

Vastly outnumbered by the forces arrayed against them, the militants are fighting back with suicide car bombs and snipers embedded among hundreds of thousands of civilians they are effectively holding hostage.

Conditions in the shrinking area under militant control are increasingly desperate as civilians resort to eating weeds and many are killed under heavy bombardment.

The number of people fleeing Mosul has more than doubled to about 10,000 a day since Friday, according to Iraqi government figures.

Defense analyst and former general Jasim al-Bahadli said the strategy adopted by Iraqi commanders was to splinter the remaining militants into smaller groups and attack them on multiple fronts to disrupt their command and control.

'By taking back all the districts surrounding the Old City, the militants will have no chance to receive any back-up or reinforcements,' he said.