The U.S.-led coalition and Iraqi forces are battling to retake the city from the terror group. Iraqi security tried to enter the city at dawn, a senior member of its rank told CNN. The official requested anonymity because he was not authorized to speak to the media.
ISIS fighters used suicide car bombs, RPGs and snipers to beat back the Iraqi forces, the official said. The Iraqi side suffered losses, the official added, declining to say how many.
The most intense fighting took place in Nuaimiya, a farming area considered to be one of the last strips of land that separated Iraqi forces from ISIS south of Falluja.
ISIS claimed that its fighters stopped Iraqi security forces and prevented them from advancing toward southern Falluja. The group said it killed at least 25 Iraqi troops and destroyed six army vehicles. CNN cannot confirm that.
In a video released by Iraq's Ministry of Defense, Iraqi Gen. Hamid al-Maliki announced Monday the city was encircled by Iraqi troops.
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.
The next step in the military's plan was to push into the heart of the city and drive ISIS out, which may be the most dangerous part of the operation.
Meanwhile, the danger for innocent people caught in the middle intensifies. There are reports that hundreds of families are being used by ISIS as human shields in the center of Falluja, said Caroline Gluk, spokeswoman for the United Nations refugee agency in Baghdad.
The agency has also received reports of civilian casualties, deaths caused by heavy shelling, she said.
About 3,700 people -- or more than 600 families -- have fled Falluja over the past week, since the offensive to retake the city from ISIS began, according to the UNHCR.
Tens of thousands in crossfire
Tens of thousands of people 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 United Nations.
Maliki says aerial forces have provided cover for ground operations, helping to take out ISIS fighters and destroy their equipment.
"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.
Falluja and Mosul are the last two Iraqi cities
under ISIS' control.
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 sending 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 stocks 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.
Farther north, thousands of Kurdish Peshmerga troops are involved in an offensive to retake formerly Kurdish villages near 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.