An operation began on February 19 to retake Mosul's west, the last major Islamic State group urban bastion in the country, which includes the Old City.
Iraqi forces have since retaken several neighbourhoods despite bad weather that has hampered air support.
In the ancient central district, hundreds of thousands of civilians are believed to be trapped under IS rule in a warren of densely populated, narrow streets which restrict the use of large armoured vehicles.
Related reading
Amal Clooney urges Iraq to allow probe of IS crimes
Federal police commander Lieutenant General Raed Shakir Jawdat said Friday that Iraqi forces backed by artillery and drones had advanced in the district.
"Federal police and Rapid Response units imposed their complete control over the Al-Basha Mosque... and the Bab al-Saray market in the Old City," he said.
The two sites lie on the edge of the district in the heart of Mosul, next to the Tigris river that slices the city in two.
Further west, forces from the elite Counter-Terrorism Service have pushed into the Al-Rissala and Nablus quarters, senior commander Staff Lieutenant General Abdulghani al-Assadi said.
"The situation is good," he said.
Major setback
The fall of Mosul, Iraq's second city, would be a major setback for IS following months of losses in Iraq and neighbouring Syria.
But its loss would not mean the end of IS in Iraq, according to Emily Anagnostos of the Washington-based Institute for the Study of War.
"The US and coalition will need to ensure their continued presence in Iraq after Mosul's recapture, which could occur within a month, in order to clear remaining ISIS-held areas and ensure stability in recaptured areas," she said in an online report, using another name for IS.
Mosul has huge symbolic significance for the jihadist group: it was from the Old City's Al-Nouri mosque that its secretive chief Abu Bakr al-Baghdadi declared the establishment of his "caliphate" in July 2014.
Iraqi forces backed by an international US-led coalition launched a vast, long-awaited operation in October to oust the jihadists from Mosul, completing their recapture of the east in January.
Watch: The Road to Mosul
Since launching their assault on the western sector, Iraqi forces have taken several districts and key buildings including the headquarters of Nineveh province's regional government and a railway station.
But the United Nations has warned that the exodus of tens of thousands of west Mosul residents could overwhelm aid groups trying to help them.
"The number of people is higher than expected," Lise Grande, the UN's humanitarian coordinator in Iraq, said.
"If the pace accelerates further, it's going to stretch us to the breaking point."
Iraq's Ministry of Migration and Displacement says more than 150,000 people have so far fled their homes in Mosul's west, with two-thirds taking refuge in camps near the city where they receive food, blankets and foam mattresses.
Grande said the UN was concerned for the plight of people still trapped inside the city, forced to choose between staying amid the fighting or risking being targeted by jihadists if they decide to flee.
"We fear the civilians there might be trapped in an extremely difficult situation," she said.
The Reach Initiative, a group that helps aid groups collect data on humanitarian crises, said the situation in west Mosul was "severe to extremely severe".
"In areas still under [IS control], there is no access to markets and people are surviving on depleting food and water stocks, without access to electricity, fuel and healthcare," it said.
