Fighters battling to retake Fallujah from the Islamic State of Iraq and the Levant (ISIL) group say they have secured its southern edge and have almost encircled the whole Iraqi city.

A leader of the Iran-backed Shia coalition taking part in the offensive said on Sunday the only side of Fallujah that remained to be secured by pro-Baghdad forces was part of the western bank of the Euphrates.

"We are now at the gates of Fallujah," Abu Mahdi al-Muhandis, deputy leader of the Popular Mobilisation Forces, told a news conference broadcast on state TV.

People fleeing Fallujah have been using anything that floats to help them get across the river, which is about 250 to 300 metres wide at the crossing point in farmland just south of the city, provincial council head Shakir al-Essawi said.

READ MORE: Iraqis drown while fleeing Fallujah

About 50,000 civilians live in Fallujah, 50km from Baghdad, with limited access to water, food and healthcare, according to a UN estimate.

Prime Minister Haider al-Abadi said on June 1 the offensive on Fallujah had been slowed down in order to protect civilians.

Sunni politicians have voiced concern that the presence of Shia militias alongside the army in the battle could lead to an increase in sectarian violence.

The Popular Mobilisation Forces have since the start of the operation confined their action to Fallujah's outskirts and left elite federal forces to conduct breaching operations.

But the Popular Mobilisation Forces' military commander, who goes by the name of Abu Mahdi al-Muhandis, said that could change if the fighting drags on.



"We're partners in the liberation. Our mission is not yet done," he told reporters in Baghdad.



"We have accomplished the task given to us, which was to surround [Fallujah] while the liberation was assigned to other forces.



"We are still in the area and we'll continue to support [them] if the liberation happens quickly. If they are not able, we'll enter with them."

Fallujah is a historic bastion of the fight against the US occupation of Iraq and the Shia-led authorities who took over after the overthrow of Saddam Hussein, a Sunni, in 2003.