NEW: At least seven people have been killed and 31 wounded in shelling, medical officials say
NEW: Al Qaeda-linked fighters warn Sunnis to stop working with the government
NEW: "There will be no withdrawal," Prime Minister Nuri al-Maliki says
But tensions between Sunnis and Shiites have been running high since last year when security forces raided a protest camp in Hawija. Dozens of people were killed in the raid, which was captured by local journalists.
The fighting in the Sunni province in recent days has posed a serious challenge to Prime Minister Nuri al-Maliki and his Shiite-dominated government, raising questions about his ability to hold the country together amid a rising insurgency.
Conflicting reports have the Islamic State of Iraq and Syria (ISIS), formerly the group commonly known as al Qaeda in Iraq, in partial control of Falluja, the site of some of the bloodiest fighting between U.S. forces and insurgents during the Iraq War.
The al Qaeda-backed group proclaimed it was in control of the heart of the city, with its flags planted on some government buildings and police stations, while two police officials and two tribal leaders told CNN that the vast majority of the city – including government institutions and police stations – was in the hands of Sunni tribesmen fighting alongside security forces.
At least seven people were killed and another 31 were wounded in the shelling, according to medical officials at Falluja General Hospital.
Fierce street-to-street fighting, between the insurgents and Sunni tribesmen aligned with police, also was reported in the provincial capital of Ramadi. Police say a number of casualties have been reported in the city, but did not offer a specific number.
The al Qaeda militants in Ramadi warned al-Maliki and tribesmen to withdraw from the area or face violent consequences.
In a video posted on YouTube, militants wearing masks warned al-Maliki’s supporters in Ramadi that they must stop helping “al-Maliki commit crimes in Anbar or you will face harsh punishments.”
Al-Maliki has vowed to crush the insurgency in Anbar province, where the Sunni insurgency – al Qaeda in Iraq – flourished following the 2003 U.S.-led invasion that toppled Saddam Hussein.
“There will be no withdrawal,” al-Maliki said in a speech Saturday carried by Al-Arabiya.
Dozens have been killed and more than 100 wounded in fighting that broke out late last week following a raid on the home of prominent Sunni lawmaker Ahmed al-Alwani, a prominent supporter of anti-government protests.
Al-Alwani’s brother and five of his bodyguards were killed during the raid, during which at least 16 other people were wounded, authorities said.
That was quickly followed by a move to close two protest camps that were established last year as part of ongoing demonstrations against the Shiite-led government, which protesters have said has marginalized the country’s minority Sunni population.
Since the withdrawal of U.S. troops in 2011, Iraq’s minority Sunnis have become increasingly marginalized. They have accused al-Maliki of trying to consolidate power by carrying out mass arrests of Sunnis.
Even so, a number of Sunni tribal leaders struck a deal with the government this week to fight alongside Iraq’s security forces against al Qaeda forces.
The deal the government made with the Sunni tribal fighters was comparable to a 2007 U.S. pact that saw Sunnis turn on al Qaeda, siding with American and Iraqi forces to bring about an end to the terrorism.
The fighting between Sunni militants against Shiite-dominated forces was reminiscent of fighting during the height of the Iraq war in 2006 and 2007, when sectarian violence nearly tore the country apart.
State Department spokeswoman Marie Harf dismissed suggestions that the United States has abandoned Iraq following its withdrawal from the country in 2011.
“Let’s be clear who’s responsible for the violence – It’s the terrorists who were behind it,” she told reporters in Washington on Friday. “That’s why we are partnering with the Iraqi government very closely to fight this shared threat. At the end of the day, we can certainly help them fight it, but we also want to help them build their own capability to do so themselves.”
The United States is sending weapons, including Hellfire rockets and drones, to aid in the campaign against terrorism, officials have said.
Harf said a number of American officials in Iraq and in Washington “remain in touch with all of the different parties in Iraq.”
CNN’s Neda Farshbaf and Yousuf Basil contributed to this report.