Social media carried posts showing black smoke and reports that it came from houses and cars set alight by the fighters.
Abadi said on June 22 that "we will declare a complete control over Mosul in the coming days".
But more than 100,000 civilians, of whom half are thought to be children, remain trapped with little food, water or medical treat-ment. "They regrouped and launched counter-attacks", Staff Lieutenant General Abdulwahab al-Saadi told AFP.
"Yarmuk is being searched house to house", he said, adding that two groups of ISIL attackers were still believed to be in the area.
The military has announced it has captured Faruq district on the north-western side of the Old City.
The fall of Mosul would mark the end of the Iraqi half of Islamic State group's "caliphate" as a quasi-state structure, but IS group would still hold sizeable, mainly rural and small-town tracts of both Iraq and Syria.
"From a military perspective, Daesh is finished", Lt-Gen Assadi said, using an Arabic term for ISIS.
Although Iraqi forces dislodged the insurgents from this part of the city months ago, battles are still raging on the western side across the Tigris.
In Syria, ISIL's former stronghold is almost encircled by a US-backed, Kurdish-led coalition.
Sunday's attacks targeted Hay al-Tanak, a poor neighbourhood west of the Old City, the last stronghold of the militants in Mosul.
But thousands of civilians were thought to still be trapped in the Old City in western Mosul, which is under IS control.
The Iraqi government once hoped to take Mosul by the end of 2016, but fighting has dragged on as militants have reinforced positions in civilian areas, effectively using residents as human shields.
"We can confirm the al-Nuri mosque was destroyed", the Washington Post reported the US-led global coalition as saying.
They said that the extremists had intentionally detonated bombs within the mosque and in the 800+-year-old minaret next to it. Extremists claimed that it was the US -led forces that targeted the buildings instead but footage shows the force of detonation coming from within.
IS fighters blew up the 12th century mosque in the heart of the Old City last week, along with its landmark minaret, according to the USA -led coalition and the Iraqi Ministry of Defense - an act of destruction that the authorities in Baghdad interpreted as the militants' message of defeat in the face of the relentless Iraqi offensive.