Thousands of civilians are estimated to remain inside Fallujah, located about 65 kilometres west of Baghdad, which IS has held for over two years.
On Sunday, government forces launched a large-scale offensive, teaming up with paramilitary troops and backed by aerial support from the US-led coalition.
Nearly 20 families have fled from Fallujah's outskirts, where sporadic clashes have been taking place, since the offensive started, said Shakir al-Issawi, the head of the council in the nearby town of Amiriyat al-Fallujah.
The Norwegian Refugee Council, an aid group working with refugees and the displaced in Iraq, reported that only 17 families has fled Fallujah since Sunday night and that most had fled from the city's outskirts.
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"There is no information for civilians about safe exits," said Becky Bakr Abdullah, an NRC spokeswoman.
"There is also the fear of being killed for attempting to flee," Abdullah said, explaining that multiple families said IS is threatening residents with death if they attempt to flee.
The International Organization for Migration put the number of newly displaced families at 125, about 750 individuals, in the past two days.
Karim Al-Nouri added that forces had secured three safe corridors for families to flee, but the militants blocked those outlets to prevent them from leaving.
Fallujah was the site of two bloody battles against US forces in 2004. It is part of the so-called "Islamic Caliphate" the militants declared in territories under their control in Iraq and neighboring Syria. IS still controls key areas in Iraq, including the second-largest northern city of Mosul.