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

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

But a confederation of Shia militias known as the Popular Mobilisation Forces have 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 the notorious militias units would not enter the city, amid fears of sectarian unrest and abuses against Fallujah's Sunni population.

But the militia units' 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.

"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."

Pro-government forces have almost completely surrounded the city, where Mohandis said 2,500 IS fighters are still holed up with thousands of civilians.

Backed by air strikes, Iraq's elite counter-terrorism service has over the past week tried to break into the city centre. 

They have been slowed by tough resistance and concerns over the presence there of an estimated 50,000 civilians.

Leaders and fighters of the various militia groups fighting under the umbrella of the Popular Mobilisation Front have sent mixed messages over the issue.

Hadi al-Ameri, head of the Tehran-backed Badr militia has repeatedly stressed that the forces should not enter central Fallujah.

Yet he told The Washington Post in article published on Saturday: "No one can stop us from going there."

Slow assault

Mohandis is a key figure in Iraq's battle to retake the territory lost to the militants in 2014.

He is considered one of the most powerful men in the country and has close ties to Iran.

Mohandis argued that the human and material cost would rise the longer the operation dragged on.

"A lengthy liberation operation would cost the security forces more and would inflict more destruction on the city, just like what happened in Ramadi," he said.

Ramadi is the capital of the vast Anbar province, where Fallujah is also located. It was retaken by government forces early this year after falling to IS in May 2015.

Federal forces relied heavily on air support from the US-led coalition to recapture the city, with entire areas levelled.

Asked about the PMF's role in Fallujah's liberation, the spokesman for the Joint Operations Command coordinating the fight against IS said Abadi would have to approve any change in the current plan.

"[PMF] have already done their part by liberating hundreds of square kilometres (square miles) and surrounding the militants who are inside the city," Brigadier General Yahya Rasool told AFP.

"Only the commander-in-chief of the armed forces has the authority to decide who should be involved in breaching operations."

On Sunday, elite Iraqi forces were battling IS on the southern edge of Fallujah, in the Jbeil and Shuhada neighbourhoods.

"There is some resistance by Daesh [IS], but a little less than in previous days," said Lieutenant General Abdel Wahab al-Saadi.

He added that on the northern side, police and other forces still had not reached the boundaries of the city but were nearing the train station.

According to the UN, IS militants have been using 300-400 Iraqi families as human shields in Fallujah.

UN officials have received "credible reports that families are being concentrated into the centre of the city by Daesh [IS] and they are not allowed to leave these concentration points," said UN deputy representative to Iraq Lise Grande.

"That would suggest that Daesh [IS] could be using them or may intend to use them as some kind of human shield," she told reporters.

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