In a statement, Unicef warned that "food and medicine are running out and clean water is in short supply", EFE news reported
Since the start of the military operation in Fallujah, which lies some 60 km west of Baghdad, "very few families have been able to leave. Most have moved to two camps while others have sought refuge with relatives and extended families," the statement said.
On May 23, the Iraqi army, backed by warplanes from the US-led international coalition, began a military offensive against the city.
Since then, both the Army Command and Prime Minister Haidar al-Abadi have made several appeals for people to leave the city through humanitarian corridors opened by the security forces.
However, according to Unicef, the plan to evacuate civilians has not been successful so far.
"Children face the risk of forced recruitment into the fighting, strict procedures for security screening and separation from their families," the statement explained.
"Children who are recruited see their lives and futures jeopardized as they are forced to carry and use arms, fighting in an adult war."
Unicef called on all parties at the end of the statement to protect children inside the besieged city, and "provide safe passage to those wishing to leave ... and grant safe and secure environments to displaced civilians."
The offensive against Fallujah, the second largest IS stronghold in Iraq after Mosul, entered its third phase on Sunday.
Pro-government forces have encountered stiff resistance there by the armed extremists.