Soldiers have secured a neighbourhood to the south of the city and are preparing for an assault, a general in the Iraqi government forces told reporters.
But with up to 60,000 civilians trapped between the jihadists and the elite special forces at the forefront of the Iraqi operation, those still inside Fallujah are becoming increasingly desperate.
It is difficult to verify the exact numbers of those drowned in their attempts to escape across the Euphrates, which runs adjacent to the west of the city, but the UNHCR has confirmed that several people have died in their attempt to escape.
Shakir al Essawi, head of the local provincial council, said hundreds of families were attempting the crossing and the bodies of two children, a woman and an older man had been recovered from the water.
Doctors at the al-Amariya Hospital, west of Baghdad, told The New Arab the death toll was as high as 18, including seven children and three women.
“They tried to escape from death but were swallowed up by the river,” Dr Mohammad al-Isawi said.
Dramatic pictures showed families, old and young, fleeing across the broad river with whatever possessions they could carry.
Mr Essawi said: “They are using empty refrigerators, wooden cupboards and kerosene barrels as makeshift boats to cross the river.
“It's totally unsafe and this is why innocent people are drowning.”
It is two weeks after Iraq launched its operation to recapture Fallujah, one of the last Isis strongholds in the country and second in size only to Mosul.
Lieutenant General Abdel Wahab al-Saadi said on Sunday that his forces had secured the largely agricultural southern neighbourhood of Naymiyah, under cover of US-led coalition air strikes.
The operation was announced in May, more than two years after Fallujah fell to Isis in a crushing defeat for the government.
Now well-embedded in the city, Isis has built tunnels and laid booby-traps and mines to make up for their disadvantage in terms of numbers.
The militants have also told civilians they will be killed if they try to flee or display white flags on their houses, as instructed by the Iraqi government.
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