Elite Federal Police and Rapid Response units on Monday resumed their cautious advance on the al-Nuri Mosque in western Mosul's Old City.
The Iraqi army, with the support of helicopters, advanced into the Old City of Mosul and an area near the mosque on Sunday.
Reports however indicate that up to 600,000 civilians may be caught inside the city. But thousands of people took advantage of the fog and rain in the early morning to flee Islamic State-controlled areas and reach the safety of government lines.
"It is one of the toughest areas, and in fact it is hard, chaotic and very tight, but our forces managed to get out of their vehicles and enter the Old City", Abdulla added.
Iraqi forces battling Islamic State militants in Mosul have approached the mosque where the leader of the extremist group declared its self-styled caliphate in the summer of 2014.
"The challenges are, firstly, how to avoid hitting civilians because they are using them as human shields", Juburi said.
Almost 100,000 Iraqis have fled western Mosul over the past three weeks, the International Organization for Migration (IOM) said on Wednesday.
Iraqi government forces, backed by US advisers, artillery and air support, have cleared the east and half of western Mosul and are now focused on controlling the Old City.
Iraq launched its military offensive to retake western Mosul from the Islamic State, also known as Daesh, ISIL and ISIS, on February 19.
They retook the east side of Mosul in January before setting their sights on the more densely-populated west of the city, the last major urban centre held by IS in Iraq.