Iraq's elite counterterrorism service moved to within three kilometres of central Fallujah Friday and consolidated positions in the south of the city, the operation's commander said.

Speaking to AFP from the edge of the city's Shuhada neighbourhood, Lieutenant General Abdelwahab al-Saadi said the operation to retake one of the Islamic State group's most emblematic bastions was progressing well.

"Daesh (IS) wanted the battle to take place outside the city but we have moved in, and retaken all this area in eight days," he said, standing on rooftop overlooking Fallujah's southern neighbourhoods.

"Our troops are here," he said pointing on his tablet computer to spots along one of the main streets in southern Fallujah. "That's 3.1 kilometres (less than two miles) from the main official building in the centre."

"We'll be there, in the very centre, in days. Days, not weeks," he said.

Plumes of smoke and dust rose up in Fallujah's low and grey skyline as special forces carried out controled detonations of bombs planted by IS and jets struck IS targets.

An Australian accent called in on the radio of a counterterrorism service (CTS) coordinator.

"I've got two guys in a building," it said, spelling out coordinates.

"Which one? The building facing east or west?" the CTS officer asked in perfect English before taking a sip from a can of energy drink. "West? OK, you're clear to engage."

Minutes later, a jet dropped its payload on the target and a huge mushroom of grey dust rose up.

"Splash on target," the CTS officer radioed back, confirming the strike was successful.

Iraqi forces are receiving air support from the US-led coalition that was formed after IS took over Iraq's second city Mosul and large parts of the country two years ago.

The operation to retake Fallujah, which lies only 50 kilometres (30 miles) west of Baghdad, was launched on May 22-23.

The first phase focused on sealing the siege of the city and the CTS has been leading a second phase that saw forces break into the urban centre of Fallujah.

Concern over the fate of tens of thousands of civilians has slowed Iraqi forces' advance but Saadi said the operation was meeting its goals.

"More than 500 Daesh members have already been killed since the start of the operation," he said.

"Fallujah is a very symbolic place for Daesh... but the battle is not different from other ones and when they are trapped, they try to run away just like they did before," Saadi said.

IS fighters have tried to leave the besieged city by blending in with the flow of displaced civilians who are attempting to escape by crossing the Euphrates south of Fallujah.

According to the Norwegian Refugee Council which runs displaced camps in Amriyat al-Fallujah, south of Fallujah, more than 20,000 people have fled the fighting over the past three weeks.