Iraqi forces entered Fallujah on Monday in their quest to retake the city from Daesh.
Government troops begin their attack from the south side of the city and had "notable progress," Lt. Gen. Abdul Wahab al-Saaidi said to Sumariyah News.
"Security forces from the counter-terrorism and army and ... Anbar police units began at dawn this morning a wide-scale military operation to breach the center of Fallujah city," Saaidi said.
The city includes an estimated 50,000 civilians.
Ground forces were backed by air support from the Iraqis and U.S.-led coalition.
"With God's blessing we have launched the third phase of the operation to storm the center of Fallujah 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. "There is heavy air cover for this operation from our heroes in the Iraqi air force and the coalition."
Earlier Monday, Iraqi military units and supporting militia took settlements from Daesh near Fallujah, including Saqlawiya, about 6.5 miles northwest of the city, and the villages of al-Buaziz, al-Bu Efan and al-Shiha, north and west of Fallujah, state-run TV reported.
Iraqi government troops, backed by Shia militias known as the Popular Mobilization Units and an air campaign by the coalition, began an operation more than a week ago to retake Daesh strongholds about 40 miles west of Baghdad.
The U.N. Refugee Agency says an estimated 50,000 people are at risk in the city.
"There are reports of a dramatic increase in the number of executions of men and older boys in Fallujah 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."
Several hundred people, mainly women and children, fled Fallujah on Friday.
Last week, Iraq's Prime Minister Haider al-Abadi urged residents to leave the city through safe passages set up by security or place a white sheet over their homes.
Those who left said the city had massive food shortages.
Daesh took Fallujah in January 2014. Along with Mosul, it is one of two major cities held by the Daesh. The population was more than 300,000 before it was seized by al-Qaeda and then by Daesh.
Because it is 40 miles east of Baghdad, it has used as a staging area for suicide attacks on targets inside the capital.
On Monday morning, there were two suicide bombings in the northern Baghdad neighborhoods of Shaab and Tarmiyah neighborhoods.
Daesh media claimed responsibility for the attacks, saying it killed more than 20 "apostates from the Shiite security services."
The Daesh commander in Fallujah was among 70 militants killed last week in an airstrike, the U.S. military announced.
Maher al-Bilawi was killed when the U.S.-led coalition bombed the city, one of more than 20 airstrikes over four days, Army Col. Steve Warren told reporters Friday. He said the attack was carried out two days earlier.
By Allen Cone