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. But as they close in on the ancient central district they face particular difficulties.
Hundreds of thousands of civilians are believed to be trapped under IS rule in the warren of densely populated, narrow streets which restrict the use of large armoured vehicles.
"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.
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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.