"According to the Iraqi military, the fighting is still very intense," Al Jazeera's Charles Stratford, reporting from Erbil, said.
The military said Iraqi "anti-terror" units are trying to take control of a small area that is less than 200 sq metres.
"They say there are around 100 ISIL fighters still in the area," Stratford said, adding that the Iraqi army is also trying to stop ISIL fighters who are trying to escape by swimming across the Tigris River.
The final push is being hindered because up to 20,000 civilians are still trapped in the area.
Earlier this week, the United Nations warned that civilians caught in the middle of the battle are in "extreme danger".
"The people that are still trapped inside of these pockets are in terrible condition" facing shortages of food, UN humanitarian coordinator in Iraq Lise Grande told AFP news agency.
The latest developments come a little more than a week after Iraqi Prime Minister Haider al-Abadi declared that ISIL's "state of falsehood" had ended after Iraqi forces retook al-Nuri Mosque, where the group proclaimed its self-styled caliphate three years ago.
With air support from the US-led coalition, Iraqi forces launched the battle for Mosul in October, retaking the eastern part of the city in January and starting the operation for its western part the next month.
Since then hundreds of civilians have been killed and more than 850,000 people have been displaced, according to the Iraqi government.
ISIL has lost much of the territory it once held over the last three years. Mosul is their last urban bastion in Iraq.