ISIS has stepped up the use of suicide bombers in the ongoing fight for west Mosul, sending them in waves against the Iraqi Security Forces fighting in the city's tight streets.
Civilians trapped in Mosul suffer from a dire lack of food, water and fuel, while many have limited access to hospitals.
Iraqi armed forces hope to capture the mosque - where Baghdadi announced the "caliphate" - in the next few days.
Iranian-backed Shiah paramilitary forces are fighting Islamic State in that part of the country where Baghdadi is believed to be hiding, according to U.S. and Iraqi officials.
When the operation to retake Mosul was launched last October, the United Nations estimated more than a million civilians were still living in the city.
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Since Friday, the government has been working to get civilians out of the areas in plans to attack, dropping leaflets to alert citizens to "safe passages" where they could flee with the help of "guides, protectors and (transportation)".
They say they missed their self-appointed timeframe to protect the civilian lives, a reminder that the civilian casualties is already high, especially after the US-led global coalition have taken responsibility for more than 100 lives lost in one single airstrike on March 17.
Popular Mobilization is taking part in the U.S-backed Iraqi campaign to defeat Islamic State in Mosul and the surrounding province of Nineveh.
"Police, counterterrorism and army units started storming the al-Zanjili District and headed toward the Old City", said federal police head, Lt. Gen. Ra'ed Shaker Jawdat, CNN reported.
In February, Iraqi forces closely backed by the USA -led coalition launched the operation for western Mosul, initially planning to clear it of IS fighters south-to-north.
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Iraqi forces, backed by heavy USA -led air and artillery strikes, began a new offensive into the Islamic State's final bastions in the city. They were similarly unaware that Is had placed explosives at the site, the report said.
Aid groups say they are concerned for the safety of civilians following calls from Iraq's government for residents of the Islamic State group-held Old City to flee the area immediately.
Since the Mosul operation was launched Iraqi forces have encouraged civilians to remain in their homes to avoid massive displacement, and IS fighters have repeatedly targeted fleeing civilians with small arms and mortar fire.
She told media that the residents were in grave danger because the Islamic State militants are directly targeting families in the city.
In the fight for Fallujah and Ramadi, cities that were also overrun by IS in 2014 as the group seized vast swaths of territory in Iraq, there was a tipping point in the battles - the moment when the militants' hold on a city had shrunk to only a handful of neighborhoods.
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