On Sunday, commanders announced the start of the «final chapter» of the offensive, with Counter-Terrorism Service, Army and Federal Police attacking the Old City from all directions.
The army believes that there are no more than 300 militants left in Mosul, compared with almost 6,000 at the start of the offensive in mid-October.
However, the densely-populated Old City’s narrow streets — Mosul’s historic heart — means the assault will be a major challenge, with troops having to clear the area house by house.
On Monday, Gen Maan al-Saadi of the Counter-Terrorism Service told AFP news agency that that his troops had made progress in the Farouq district, but that resistance had been «fierce».
«[Militants] have blocked every entrance, planted IEDs [improvised explosive devices] and booby-trapped houses our forces might be near,» he said.
«Penetrating was very difficult. Today the fighting is face-to-face.»
Troops stationed near the Grand Mosque of al-Nuri, where IS leader Abu Bakr al-Baghdadi proclaimed a «caliphate» in 2014, used loudspeakers to tell militants holed up inside «surrender or die».
On Sunday night, Iraqi aircraft dropped almost 500,000 leaflets over the city, urging civilians to «stay away from open places» and «exploit any opportunity» to escape.
The International Rescue Committee, an aid group that operates in Iraq, warned that this would be «a terrifying time» for civilians trapped inside the Old City.
«With its narrow and winding streets, Iraqi forces will be even more reliant on air strikes despite the difficulty in identifying civilians sheltering in buildings and the increased risk of civilians being used as human shields by [IS] fighters,» it said.
«The buildings of the old town are particularly vulnerable to collapse even if they aren’t directly targeted, which could lead to even more civilian deaths than the hundreds killed so far in air strikes across the rest of the city.»
Gen Saadi told state TV on Sunday that troops were «trying to be very careful, using only light and medium weapons» to avoid civilian casualties.
The IRC also warned that civilians attempting to flee the fighting faced significant risk of getting caught in the crossfire or being targeted by IS snipers.
The UN said last week it had received credible reports that more than 200 civilians were shot dead by militants in the Shifa district on 1 and 3 June alone.
Civilians unable to flee the Old City are meanwhile facing «desperate conditions», with little food and no clean water, according to the UN.
Since the start of the battle for Mosul, an estimated 862,000 people have been displaced. About 195,000 have since returned to their homes.