WATCH: Iraqi Troops Now Surrounding ISIS Stronghold In Falluja

FALLUJA, IRAQ — (CNN) The ISIS stronghold of Falluja has been surrounded, Iraqi General Hamid al-Maliki said in a video released by the Ministry of Defense.

Iraqi forces, in conjunction with Iraqi and coalition air support, started the operation to retake the city Monday, an Iraqi military spokesman said, capturing some villages on the outskirts of Falluja on Monday.

The next step in the military’s plan is to push into the heart of the city and drive the militants out, which may be the most dangerous part of the operation.

Al-Maliki says that aerial forces have provided cover for ground operations, helping to take out ISIS fighters and destroy their equipment.

Still, tens of thousands are at risk once door-to-door combat begins in the city, and there are reports of ISIS executing men and boys who refuse to fight for them, according to the U.N.

“With every moment that passes, their need for safe exits becomes more critical,” said Nasr Muflahi, the Norwegian Refugee Council country director in Iraq. “Getting there in the first place is near impossible for those in the city center.”

And if the city is liberated, Iraqi authorities will face a whole new set of challenges — among them, addressing the social and political problems that led to the rise of ISIS, as well as security issues.

Storm the city

The newest phase of the operation to retake Falluja was announced early Monday morning.

“With God’s blessing we have launched the third phase of the operation to storm the center of Falluja city — by our heroes in the counterterrorism forces, units of the Iraqi army and Anbar police,” Iraqi military spokesman Yahya Rasoul said on Iraqi state TV.

Iraqi forces retook the village of Nuaimiya, just south of Falluja, closing in on the city itself, al-Iraqi TV reported.

Earlier Monday, Iraqi military units and supporting militia captured a handful of settlements from ISIS near Falluja, including the town of Saqlawiya, about 10 kilometers (6.5 miles) northwest of the city, and the villages of al-Buaziz, al-Bu Efan and al-Shiha, north and west of Falluja, state-run TV reported.

Iraqi government troops, backed by Shiite militias known as the Popular Mobilization Units and an air campaign by the U.S.-led anti-ISIS coalition, launched the offensive last week to retake the ISIS stronghold, about 65 kilometers (40 miles) west of Baghdad, first targeting outlying settlements.

Capturing Karma, about 16 kilometers (10 miles) northeast of Falluja on Thursday, brought most of the territory east of the city under government control.

More than 5,000 Americans are providing training, advice and assistance to Iraqi forces and flowing equipment to them, United States Col. Steve Warren, a spokesman for the coalition, said on CNN’s “New Day” Monday.

Iraq was “invaded by these animals” of ISIS, he said, but the terror group’s grip on territory is slipping in both Iraq and Syria, he said.

Tens of thousands at risk

The U.N. refugee agency has warned that an estimated 50,000 people are at risk, caught in the city turned combat zone as the Iraqi military’s bombardment has intensified.

“There are reports of a dramatic increase in the number of executions of men and older boys in Falluja (who are) refusing to fight on behalf of extremist forces,” said Leila Jane Nassif, the U.N. agency’s assistant representative in Iraq.

“And many people have been killed or buried alive under the rubble of their homes in the course of ongoing military operations.”

Hundreds, mostly women and children, fled Falluja on Friday as Iraqi soldiers attacked to drive ISIS from the city, the Iraqi military said.

Security forces evacuated about 760 people who escaped from eastern and southeastern Falluja, the military said.

“Food is scarce in the city,” said Um Ahmed, a 40-year-old living in Falluja with her family. “We have mostly been relying on dates for our meals.”

The Norwegian Refugee Council reports the residents of Falluja have been faced with food shortages, lack of electricity and hospitals have depleted supplies of medical supplies.

“We are working around the clock in the displacement camps to provide water and emergency food rations to people who manage to flee the violence,” Muflahi, the Iraq country director for the NRC, said.

Multiple offensives

Farther north, thousands of Kurdish Peshmerga troops are involved in an offensive to retake formerly Kurdish villages near the ISIS-held town of Mosul, Kurdish officials say.

The Peshmerga-led ground offensive, backed by coalition air support, was launched early Sunday to recapture villages near Khazir, east of Mosul.

The move comes ahead of a joint offensive by Kurdish forces and Iraqi troops to take back Mosul, Kurdish media said.

Across the border in Syria, a U.S.-backed alliance of Kurdish and Arab forces last week pushed into territory north of Raqqa, the de facto capital of ISIS.

Sarmad al-Jilane, an activist with the monitoring group Raqqa is Being Slaughtered Silently, confirmed reports that ISIS has let some residents of that city flee to the surrounding countryside or Deir Ezzor as Kurdish and Arab forces pushed forward.