FALLUJAH - Iraqi forces battled the Islamic State group Monday in the opening stages of an operation to retake Fallujah, one of the toughest targets yet in Baghdad's war against the militants.

As Iraqi forces struck targets in and around the militant bastion, which saw deadly battles in 2004 between insurgents and American forces, IS claimed bombings in neighbouring Syria that killed at least 148 people.

The militant group has increasingly turned to its traditional tactic of killing civilians in bombings as it faces battlefield losses, and spokesman Abu Mohammed al-Adnani appeared to acknowledge in a recent statement that IS would probably lose more ground.

"In the early hours of the morning today, the heroic fighters advanced from different sides" to retake "all the areas occupied by (IS) around Fallujah", Iraqi Prime Minister Haider al-Abadi announced Monday in televised remarks. Abadi said the operation was supposed to start earlier, but "political problems and also the events... threatening security inside Baghdad delayed some of the preparations".

Iraq has been hit by a months-long political crisis that has paralysed the legislature, and demonstrators have twice broken into the fortified Green Zone area, storming parliament and Abadi's office.

IS has also carried out a series of deadly attacks in and around Baghdad this month.

Iraqi forces had not yet entered the Anbar province city just 50 kilometres (30 miles) from Baghdad, but an AFP photographer near Fallujah said they were advancing as aircraft hit targets inside it.

Military sources said early efforts had isolated Fallujah from Karma, an area to the northwest where IS is still present. IS, meanwhile, issued a statement claiming it had repelled "a wide attack" by Iraqi forces and destroyed multiple tanks and bulldozers.

Abadi's announcement settled the issue of which IS-held city Iraq should seek to retake next - a subject of debate among Iraqi officials and international forces helping Baghdad battle the militants.

Iraq's second city Mosul was the US military's recommended target, but powerful militias may have helped force the issue by deploying reinforcements to the Fallujah area in preparation for an assault.

On Sunday, Iraq's Joint Operations Command warned civilians still in Fallujah - estimated to number in the tens of thousands - to leave. It warned families that could not depart to raise a white flag over their location and stay away from IS headquarters and gatherings.

Officials said several dozen families had fled the city, but IS has sought to prevent civilians from leaving, as have forces on the government side, according to the UN Refugee Agency.

"Aid has not reached Fallujah since the government recaptured the nearby city of Ramadi... with the supply routes cut off by the Iraqi forces and armed groups, preventing civilians from leaving," UNHCR said.

Anti-government fighters seized Fallujah in January 2014 after security forces withdrew during unrest sparked by the government's destruction of a protest camp, and the city later became one of IS's main strongholds.

Fallujah and Mosul, the capital of the northern province of Nineveh, are the last two major cities IS holds in Iraq. Iraqi forces have regained significant ground in Anbar province in recent months, but as the militants are pushed back they are stepping up their deadly bombings.

On Monday, seven near-simultaneous explosions targeted civilian sites in the Syrian seaside cities of Jableh and Tartus, killing at least 148.

The militants have also struck Baghdad, and attacks in and around the Iraqi capital have killed more than 160 people this month.

Fallujah has a long history as an insurgent bastion, and US forces launched two major assaults on the city in 2004, in which they saw some of their heaviest fighting since the Vietnam War.

Iraqi forces have the advantage of greater knowledge of the area, especially if they employ pro-government Anbar tribal fighters in the battle, but they lack the training and enormous firepower that American forces could bring to bear.

IS overran large areas north and west of Baghdad in June 2014, and Iraqi forces performed dismally during the initial offensive despite significantly outnumbering the militants.

But the "caliphate" the militant group subsequently proclaimed has been shrinking as anti-IS forces score major victories in both Iraq and Syria, where the group had also seized significant territory.

Several key IS leaders, including its number two and the top military commander for Anbar, have been killed in recent air strikes by the US-led coalition.

Coalition spokesman Steve Warren said on social media that 21 strikes had been carried out on IS targets in and around Fallujah since May 17.