"We are concerned over the protection of children in the face of extreme violence," UNICEF Representative in Iraq Peter Hawkins said in a statement.
"Children face the risk of forced recruitment into the fighting" inside the besieged city, and "separation from their families" if they manage to leave, he added, Reuters reported.
Backed by volunteer forces and airstrikes from the US-led coalition, the Iraqi armed forces launched on May 23 an offensive to recapture Fallujah, 50 kms (32 miles) west of Baghdad.
The assault on Fallujah has begun what is expected to be one of the biggest battles ever fought against Daesh.
Fallujah was the first Iraqi city that fell under control of the ultra-hardline militants, in January 2014.
About 50,000 civilians remain in the city, according to the United Nations.
Iraqi security forces operating in Fallujah separate systematically men and boys over 12 from the families to probe possible links with Daesh.
"UNICEF calls on all parties to protect children inside Fallujah, provide safe passage to those wishing to leave the city and grant safe and secure environment to civilians who fled Fallujah," Hawkins said.