Iraq is investigating allegations that Shia militiamen helping the army retake Falluja have executed dozens of Sunni Muslim men from the city held by Islamic State group (IS).

The authorities "are following up on the violations and a number of arrests have been made," government spokesman Saad al-Hadithi said on Monday after a regional governor said 49 Sunni men had been executed after surrendering to a Shia faction.

Suhaib al-Rawi, governor of Anbar province where Falluja is located, said on Sunday that 643 men had gone missing between June 3 and June 5, and "all the surviving detainees were subjected to severe and collective torture by various means."

The participation of militias in the battle of Falluja, just west of Baghdad, alongside the Iraqi army had already raised fears of sectarian killings.

Falluja is a historic bastion of the Sunni insurgency against U.S. forces that toppled Saddam Hussein, a Sunni, in 2003, and the Shia-led governments that followed.

"Strict orders were issued to protect the civilians," government spokesman Hadithi said, adding that these instructions were also given to the Hashid Shaabi, or Popular Mobilisation Forces, the coalition of mostly Shia militias backed by Iran which are involved in the fighting.

The United Nations said last week it knew of "extremely distressing, credible reports" of men and boys being abused by armed groups working with security forces after fleeing Falluja.

Iraqi authorities routinely separate males aged over 15 from their families when they manage to escape Falluja, to screen them to ensure they do not pose a security risk and check if they may have been involved in war crimes.

U.N. human rights chief Zeid Ra'ad al Hussein said that screening was legitimate but should not be done by paramilitary groups.

Civilians from Falluja had suffered "two and a half years of living hell" under Islamic State and faced not just enormous danger in escaping but also "double jeopardy in the form of serious human rights violations," Zeid said.

A spokesman for the U.S.-led coalition fighting Islamic State said the Baghdad government was aware of the abuses.

"We know that the prime minister has come out and said that he believes that these abuses have happened and that he ... has demanded accountability of any perpetrators," Colonel Chris Garver said. "We think that is the right course of action."

The Iraqi army launched the offensive on Falluja on May 23, with air support from the U.S.-led coalition. The United Nations has said up to 90,000 people are trapped in the city with little food or water.

Repeated phone calls to three spokesmen of the Popular Mobilisation Forces were not answered. Last week, one of them, Kareem Nuri, said past accusations of human rights violations were "politically motivated and baseless".