The Pentagon said meanwhile that US Apache helicopters hit an Islamic State group target for the first time in Iraq, in the north of the country.
More than 500 suspected IS members have been arrested trying to sneak out with civilians since forces ramped up efforts to retake Fallujah, one of the group's most emblematic bastions, two weeks ago.
"We have arrested 546 suspected terrorists who had fled by taking advantage of the movements of displaced families over the past two weeks," Hadi Rzayej, the police chief for Anbar province in which Fallujah is located, said yesterday.
When civilians reach government forces, teenage boys and adult men are screened separately. Some are released after a few hours while others undergo more thorough interrogation.
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"Daesh (IS) is fleeing among the civilians, we have arrested many and are investigating the suspects," said Abdelwahab al-Saadi, the overall commander of the operation.
Until last week, an estimated 50,000 civilians were still trapped in the centre of the city, which is one of the jihadist group's last bastions in Iraq and lies only 50 kilometres west of Baghdad.
UN deputy representative to Iraq Lise Grande said yesterday that more than 7,000 people have fled Fallujah in recent days using the safe corridor set up by Iraqi forces, who were working on opening a second protected route.
Yet the flow of residents fleeing via the corridor and through the Al-Salam intersection to the southwest of the city appeared to dry up on Monday, the Norwegian Refugee Council's regional media adviser Karl Schembri said.
He said nonetheless that more than 2,600 new arrivals had been recorded in displacement camps yesterday, mostly civilians from the outskirts of the city.
Estimates for the number of IS fighters holed up in Fallujah vary from 1,000 to 2,500.