The new progress began on Saturday morning when the Iraqi army pushed into al-Shifaa neighbourhood, the police and elite Rapid Response forces advanced into the neighbouring al-Zanjili area, and the commandos of the Counter-Terrorism Service pushed into the IS-held part of al-Seha, Xinhua news agency quoted a military official as saying.
"Our forces are capable of winning the fight of the Old City and regaining the al-Nouri Mosque, and ISIS are losing control of the vital area close by al-Hadba minaret (adjacent to al-Nouri Mosque)".
The Islamic State is putting women on the front line in an effort to hold back Iraqi forces from taking the group's last major stronghold in western Mosul.
Several residents of the Mosul neighborhood hit by a USA airstrike in March are denying the findings from the coalition's probe of the attack. They spoke on condition of anonymity in line with regulations.
Iraqi forces launched a major operation to retake Mosul in October past year, fighting their way to the city and retaking its eastern side before setting their sights on its smaller but more densely populated west.
All the bridges spanning the Tigris River, which roughly divides Mosul into its eastern and western half, has been destroyed by coalition airstrikes, but residents reported IS fighters still managed to flee west, ahead of the Iraqi advances.
The offensive is the latest push in the more than seven-month battle to retake Mosul, a lynchpin in IS' now crumbling bid to establish a cross-border jihadi "state".
This is the opposite of the strategy Iraqi forces employed in east Mosul, where they urged civilians to stay in their homes.
During a Sunday interview, US Defense Secretary Jim Mattis said that they were doing "everything humanly possible" to avoid civilian casualties, but in this kind of asymmetrical conflict civilian deaths are "a fact of life".
The push inside the Old City coincides with the start of the holy fasting month of Ramadan.
The fall of Mosul would, in effect, mark the end of the Iraqi half of the "caliphate" declared almost three years ago by Daesh leader Abu Bakr al-Baghdadi in a speech at the mosque.
United Nations and humanitarian officials estimate that 700,000 people - one-third of the city's prewar population - have already fled the fighting, with some 200,000 still thought to be trapped.
More than 100,000 civilians are estimated to still be inside IS-held Mosul neighborhoods.
Nasiri, who had been an adviser for Popular Mobilization Units in Iraq, also known as the Hashd al-Shaabi, was killed in a roadside bombing near Baaj, west of Mosul, Ibrahim said.