Iraqi forces launched the operation to retake Mosul in October, advancing on the city and recapturing its eastern side before setting their sights on the smaller but more densely populated west.
UN aid chief Stephen O'Brien said he was "deeply concerned" for the safety of civilians living under IS control, citing reports of families being shut inside booby-trapped homes and snipers targeting children.
Nearly 760,000 people have fled Mosul during the seven-month government offensive and UN officials say they do not know for sure how many civilians are still living in IS-controlled areas.
On Thursday, the Iraqi government urged civilians inside the Old City and surrounding districts under IS rule to leave and head across the front lines to government-controlled areas.
"As many as 200,000 additional people may try to leave in the coming days," said O'Brien in a statement.
"Those who choose to flee their homes to access assistance must be free to do so without hindrance," he said.
UN agencies have set up camps near Mosul to house those displaced by the fighting and last summer estimated that up to one million people could be forced from their homes.
The drive to retake Mosul has been supported by a campaign of US-led coalition air strikes in and around the city.
A US strike targeting IS fighters in a Mosul building in March killed at least 105 civilians when the blast caused IS weapons to explode, the US defense department said this week.
Iraqi and coalition forces did not know civilians were in the building and ultimately a strike was called in, US General Matt Isler said.
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