Iraq said Monday it had launched an investigation into possible human rights abuses against civilians fleeing the Islamic State-held city of Fallujah as an aid group said more than 4,000 more people had left the city over the weekend.

Iraqi government spokesman Saad al-Hadithi said some fighters suspected of violating human rights during the three-week-old operation to retake the city have been arrested over the past few days and are under investigation.

He did not provide details on whether the fighters are from the army or government-sanctioned paramilitary forces, which are mainly made up of Shiite militias.

Prime Minister Haider al-Abadi "issued arrest warrants against some suspects who took part in the operation a few days ago," al-Hadithi said. "The Prime Minister is closely following the investigation and we are still awaiting the results."

Local Sunni officials and human rights groups have accused Shiite militias of arresting, torturing and killing Sunnis who fled Fallujah and its outskirts.

The governor of Anbar province, where Fallujah is located, told a press conference on Sunday that 49 civilians were killed and 643 others have gone missing. Suhaib Al-Rawi described them as displaced persons from Fallujah who reached areas controlled by the paramilitary forces.

The spokesman for Iraq's paramilitary forces, Karim al-Nouri, confirmed the arrest of one fighter for alleged human rights violations, but did not provide further details. He said the accusations of abuse by paramilitary forces were "mere lies aimed at distracting attention from victories on the ground."

He said that all displaced people detained by paramilitary forces for security screening were handed over to authorities.

Last week, the U.N. human rights chief said there were "credible reports" that Iraqis fleeing Fallujah were facing physical abuse as they escaped the city held by Islamic State militants.

An estimated 50,000 civilians remain trapped inside Fallujah, located about 40 miles (65 kilometers) west of Baghdad. Their presence, along with roadside bombs, has slowed down the Iraqi forces' operation to recapture the city.

Fallujah has been under IS control for over two years and is the last major city in western Iraq still held by the extremist group. The militants have threatened anyone who attempts to flee with death and last week reportedly shot at a group of civilians attempting to flee across the Euphrates River.

The Norwegian Refugee Council said the 4,000 who fled since Saturday bring the total number of residents who escaped Fallujah since the offensive started in late May to 27,580.

Thousands more are expected to take the risky journey in the coming hours, NRC said, adding that some refugees reported IS militants were demanding 150,000 Iraqi Dinars, or around $130, from each person who wanted to leave.

The aid group warned that humanitarian resources are running low and called for at least $10 million to fund a six-month supply of water, food and basic necessities.

"Thousands of others remain trapped inside and the most vulnerable will need urgent assistance," said Nasr Muflahi, the NCR country director in Iraq.

On Sunday, the Iraqi command announced that key areas to the west of Fallujah had been taken and that Iraqi forces pushed deeper into the city from its southern edges.

U.S.-trained Iraqi counterterrorism forces, wary of coming street battles in the city, are already facing fierce resistance on the outskirts from well-entrenched militants.