"In a few days our forces will reach Cornishe Street and bring the battle to its conclusion", said Jawdat, adding that federal police had forced militants out of Ziwani mosque in the Old City's southwestern corner.
The army recaptured eastern Mosul in January and is now fighting to retake control of the western part of the city.
Hadarat Al-Saada and Al-Ahmadiyya are northwest of the historic Grand Al-Nuri Mosque, which the militants destroyed last week.
In Syria, the Islamic State-held city of Raqqa is almost encircled by a US-backed, Kurdish-led coalition.
A US-led worldwide coalition is providing air and ground support in the eight-month-old offensive.
Up to 350 militants are said to be dug in there among civilians in wrecked houses and crumbling infrastructure.
They were making extensive use of booby traps, suicide bombers and sniper fire to slow the advance of Iraqi forces from the west, the north and the south.
An estimated 862,000 people have been displaced from Mosul ever since the battle to retake the city began nine months ago.
Iraqi troops Monday captured the Al-Faruq quarter, facing the mosque, the military said.
Fighting is now focused on Mosul, ISIS's de facto capital, and Khodari claimed that the suicide attacks on Sunday "show that they will soon be defeated in Mosul".
About 850,000 people, more than a third of Mosul's pre-war population, have fled, seeking refuge with relatives or in camps, according to aid groups.
Iraqi army soldiers and volunteer fighters from the Popular Mobilization Units, commonly known by their Arabic name, Hashd al-Sha'abi, have made sweeping gains against Daesh since launching the Mosul operation on October 17, 2016.
Bernard Smith has more from the nearby city of Erbil. The extremist group is using the civilians as human shields.