Humanitarian camps are prepared for people fleeing the fighting, but thousands remain stuck in the northern Iraqi city.
Fighting has been reported in one eastern district of Mosul even though Iraqi forces recently recaptured the area from the self-proclaimed Islamic State.
Civilians have been leaving the Mithaq neighbourhood carrying belongings amid the sound of gunfire in the distance.
Most have left the east but a growing number are escaping the besieged west, still controlled by the militants.
“Some are lucky and come through, it’s question of chance, but there are those who are caught and others are stabbed – every person has a different destiny,” said one Iraqi man named as Jamal, who succeeded in fleeing Mosul’s western side.
As the second phase of the army offensive against ISIL continues, many civilians have have to wait to board buses at Bartella, to be driven to nearby camps.
A spokesman said they had first been taken to security screening points where they had received medical treatment and meals.
A new camp has been opened at Khazer for people displaced from Mosul.
The UN says at least 114,000 people have left their homes since military operations began.
The camp was built in anticipation of thousands of residents fleeing ISIL. As many as a million are said to be out of reach of humanitarian assistance.
But those arriving at Khazer are in a minority. So far most of the city’s residents have either chosen to stay or have been unable to escape.
That has worked in ISIL’s favour, slowing the advance of Iraqi forces seeking to avoid civilian casualties.