The UN's envoy to Iraq has warned of a "critical humanitarian situation" in Anbar province, as troops prepare to launch a major assault against al-Qaeda-linked militants in Fallujah.
Nickolay Mladenov said more than 5,000 families had fled the city as food, water and medicine began to run out.
An Iraqi Red Crescent official put the figure far higher - at 13,000 families.
Earlier, Iraq's prime minister urged the militants who have seized control of Fallujah to give up their struggle.
Nouri Maliki appeared to hold out the offer of a pardon if they did so.
Markets reopen
Sunni militants
allied to the al-Qaeda-affiliated Islamic State in Iraq and Syria (ISIS), as well as armed tribesmen angry with the government, overran Fallujah last week after clashes triggered by a raid on protest camps in the city and in the provincial capital, Ramadi.
On Wednesday morning, security officials and tribal leaders said Mr Maliki had agreed to delay any assault to give people in Fallujah time to persuade or force ISIS out.
"We've done our part of the deal. Now they should do theirs. If not, a quick offensive is coming," a special forces officer told the Reuters news agency.
On Wednesday the Associated Press reported that calls had been broadcast on mosque loudspeakers calling on militants to leave the city and on fleeing families to come back. Markets had also reopened, and civilian vehicles and traffic policemen had returned to the streets, it added.
But the UN's special representative to Iraq warned that food and water supplies were beginning to run out and that the humanitarian situation was "likely to worsen as operations continue".
"The UN agencies are working to identify the needs of the population and prepare medical supplies, food and non-food items for distribution if safe passage can be ensured. This remains a primary challenge," Mr Mladenov said in a statement.
According to a preliminary UN assessment, more than 5,000 families have sought refuge in the neighbouring provinces of Karbala, Salahuddin and Baghdad, but Iraqi Red Crescent official Mohammed al-Khuzaie said as many as 13,000 families had fled, and that it had provided humanitarian assistance to more than 8,000.
Fighting continued on the outskirts of Ramadi on Wednesday, although security forces backed by pro-government tribesmen largely regained control of the city two days ago.
'Sacred war'
In his weekly televised address on Wednesday, Mr Maliki vowed to continue the "sacred war" against ISIS and regain full control of Fallujah and Ramadi.
"The war that is being fought by the Iraqi security forces, tribes and all segments of Iraqi society against al-Qaeda and its affiliates is a sacred war," he said. "I call on those who were lured to be part of the terrorism machine led by al-Qaeda to return to reason."
In return, the prime minister promised to "open a new page to settle their cases so that they won't be fuel for the war that is led by al-Qaeda".
Islamist militants have capitalised on deep-seated grievances among Iraq's minority Sunni Arab community, which has complained for years that it is being marginalised by Mr Maliki's Shia-led government and unfairly targeted by the security forces.
Correspondents say the prime minister's drive to restore control of the two predominantly Sunni cities of Fallujah and Ramadi is being seen by many Sunnis as an attempt at domination and oppression, and is taking Iraq back to the brink of a sectarian civil war.
Courtesy: BBC
