The UN has said it is very concerned about the fate of some 50,000 civilians still in Falluja as Iraqi forces battle the group calling itself Islamic State.
UN spokesman Stephane Dujarric said the civilians were at “great risk” and called for the provision of safe corridors to allow them to leave.
Soldiers, police and militiamen began an offensive to retake Falluja, 50km (30 miles) west of Baghdad, on Monday.
The city has been held by IS longer than any other in Iraq or Syria.
The jihadist group overran Falluja in January 2014, six months before it routed the Iraqi army and seized control of large parts of northern and western Iraq.
Although the offensive to recapture Falluja was launched on Monday, Iraqi government forces have besieged the city and its suburbs for several months.
The UN says there have been no deliveries of aid since IS militants were driven from the nearby city of Ramadi in December and supply routes were cut.
The estimated 10,000 families living in Falluja have faced acute shortages of food, medicine and other essential items. Prices have escalated dramatically.
On Sunday, the Iraqi military urged all civilians to leave Falluja and instructed those who unable to raise a white flag over their location and stay away from IS positions and gatherings.
Speaking to reporters in New York on Monday, Mr Dujarric said: “We’re very concerned about the fate of the civilians that remain in Fallujah as the military operations are undertaken.”
Analysis: Jim Muir, BBC News, Baghdad
Falluja has been attacked many times and bombed and shelled almost incessantly since it fell into the militants’ hands in January 2014. It has withstood all that, despite huge destruction and many casualties.
Now the government has committed itself to “liberating” the city once and for all, in an operation codenamed “Break Terrorism”.
But there are conflicting assessments of how tough the battle will be.
Some believe that IS has taken such a pounding in the town that its ability to resist has been sapped. Others, in touch with sources inside the beleaguered city, say the militants have long been preparing to face such an offensive and have deployed their full array of defences, including many roadside bombs and booby traps.
“The humanitarian situation obviously remains very fluid as the fighting is ongoing.”
Mr Dujarric said civilians would “come under great danger as they try to flee” and that it was important that there were some safe corridors that they could use.
UNHCR spokeswoman Leila Jane Nassif said 80 families were believed to have fled Falluja since Friday, some of them by crossing agricultural land that lies between the city’s southern outskirts and the town of Amariyat al-Falluja.
“In some cases, their escape was at the cost of lost lives including women and children,” she added.
Ms Nassif also expressed concern for the safety of men and older boys who survived the escape. She said they were were being separated from the women and other children by security forces and taken for security screening at Habbaniya military base.
In the event that civilians are able to leave Falluja, the UN and other humanitarian organisations have set up shelters for them in nearby towns.
Iraqi Prime Minister Haider al-Abadi said on Monday that government forces had made better progress than expected.
A general in charge of the operation said they already retaken Karma, a small town that lies in a rural area north-east of Falluja.
IS meanwhile said it had repelled an attack and destroyed army vehicles.