At least 21 people, some of them women and children, were killed in Fallujah, the Wall Street Journal reported.
The militants describe themselves as supporters of the Islamic State in Iraq and al-Shams, or ISIS, the Journal said. They are Sunni Muslims who oppose the government in the Shiite-majority country.
Officials said that at the end of the day about 500 militants still controlled most of Fallujah, while about 60 percent of Ramadi remained in the hands of ISIS fighters.
Fallujah, 43 miles west of Baghdad, was the scene of two major battles during the U.S. occupation of Iraq. The first, in April 2004, began after four U.S. contractors were killed there.
The latest fighting began after Prime Minister Nouri al-Maliki ordered a crackdown on protest camps in Fallujah and Ramadi, the New York Times said. It reflects both spillover from the civil war in Syria, which has drawn militant Sunni fighters to the country, and growing Sunni dissatisfaction with Maliki's government, the newspaper said.
2014: The Year in Fashion [PHOTOS]
Syrian Al Qaida group executes Lebanese soldier