BAGHDAD - Fighters battling to retake Falluja from Islamic State said on Sunday they had the Iraqi city almost completely encircled, as civilians risked their lives trying to flee.

At least four people drowned and nine were missing after trying to cross the Euphrates river to escape the Islamic State stronghold city just west of Baghdad, medics and officials said.

The drowned bodies of two children, a women and an older man were taken to a hospital in Ameriyat Falluja, a town downstream under government control, a local official said. Police said they were looking for nine other people believed to have been on the same boat.

About 50,000 civilians live in Falluja, 50 km (32 miles) from Baghdad, with limited access to water, food and healthcare, according to a UN estimate.

Iraqi forces, with backing from Shia militias and air support from the US-led coalition, launched an offensive on May 23 to retake the Sunni city, the first to fall to Islamic State in Iraq, in January 2014.

A leader of the Iran-backed Shia coalition taking part in the offensive said the only side of Falluja 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 Falluja," Abu Mahdi al-Muhandis, deputy leader of the Popular Mobilization Force, told a news conference broadcast on state TV.

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

"They are using empty refrigerators, wooden cupboards and kerosene barrels as makeshift boats to cross the river," Essawi told Reuters. "It's totally unsafe and this is why innocent people are drowning."

Pictures broadcast by Dubai-based Arabiya TV in recent days showed families using rafts and sometimes just swimming across. Young men were seen helping old men and women disembark when reaching the opposite side and carrying them onto the bank.

Some of those who were trying to reach the river were killed by sniper fire coming from the lines held by the militants, or by explosive devices planted along the roads, said Jassim Alwan, a police captain in Ameriyat Falluja.  Essawi said more than one thousand families had managed to cross the river.

Prime Minister Haider al-Abadi said on June 1 the offensive on Falluja 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.  Falluja is a historic bastion of the insurgency against the US occupation of Iraq and the Shia-led authorities who took over after the overthrow of Saddam Hussein, a Sunni, in 2003.

Meanwhile, an Iraqi paramilitary organisation dominated by Tehran-backed militias is willing to send forces inside Fallujah if efforts to retake the city are too slow, its top commander said on Sunday.

Iraqi forces launched a vast offensive on May 22-23 against Fallujah, which lies only 50 kilometres (30 miles) west of Baghdad and is one of the Islamic State group's main bastions.

The Hashed al-Shaabi militias have since the start of the operation confined their action to Fallujah's outskirts and left elite federal forces to conduct breaching operations. Prime Minister Haider al-Abadi has stated clearly that Hashed forces would not enter the city, amid fears of sectarian unrest and abuses against the city's Sunni population. But Hashed al-Shaabi's military commander, who goes by the name Abu Mahdi al-Mohandis, 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," Mohandis said.

"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." Hashed and other forces have almost completely surrounded the city, where Mohandis said 2,500 IS fighters are still holed up.

Backed by US-led air strikes, Iraq's elite counter-terrorism service has over the past week attempted to break into the centre of the city but has been slowed by tough resistance, as well as concerns over the presence in central Fallujah of an estimated 50,000 civilians.