Saudi Minister of Foreign Affairs Prince Saud al-Faisal bin Abdulaziz al-Saud, right, listens as Secretary of State John Kerry makes a statement to the press at King Khalid International Airport, on January 5, 2014, in Riyadh. (BRENDAN SMIALOWSKI/AFP/Getty Images)

Secretary of State John F. Kerry said Sunday that the United States is ready to help Iraq in any way possible as that country began a major offensive to wrest control of two cities from al-Qaeda-linked militants. But he made it clear that no American troops would be sent in.

Kerry described the militants from the Islamic State of Iraq and Syria, or ISIS, as “the most dangerous players” in the region. But as Iraqi forces launched airstrikes and clashed with the militants in western Anbar province on Sunday, Kerry said it was Iraq’s battle to fight.

ISIS, formerly known as al-Qaeda in Iraq but renamed to reflect the group’s growing ambitions, has been extending its influence across Syria, Iraq and Lebanon. It is suffering a backlash in Syria, where it lost ground to rival rebel fighters on Sunday. But the Sunni militants’ gains in Iraq present a critical test for the Shiite-led government of Prime Minister Nouri al-Maliki.

They also leave the Obama administration worried about the renewed force of a militant movement once declared all but vanquished in Iraq.

A string of bombings in the Iraqi capital killed at least 20 people Sunday; although no one asserted responsibility, the attacks appeared linked to the fighting in Anbar. In that strategic province, meanwhile, the militants linked to al-Qaeda fought to retain their grip on the cities of Ramadi and Fallujah, which they seized last week .

1 of 11 Full Screen Autoplay Close Skip Ad × Iraqi forces battle for Fallujah View Photos Iraqi government forces have begun a major offensive to retake the militant-held Fallujah in western Anbar province. Caption Iraqi government forces have begun a major offensive to retake the militant-held Fallujah in western Anbar province. Jan. 5, 2014 Burned-out trucks litter the side of the main highway leading west out of Baghdad to Fallujah, Iraq. The trucks were destroyed during fighting the day before between Iraqi army forces and militants. Sadam El-Mehmedy/AFP/Getty Images Buy Photo Wait 1 second to continue.

The Iraqi army had encircled Fallujah on Sunday, poised for an assault. Thousands of residents fled, fearing an onslaught similar to the U.S. military’s 2004 battle for the city, then held by Sunni insurgents. It was the most deadly confrontation of the Iraq war for U.S. forces and some of their bloodiest fighting since the Vietnam War.

Meanwhile, in Ramadi, airstrikes killed 60 ISIS militants late Saturday, Iraq’s army chief, Lt. Gen. Ali Ghaidan Majid, told the National Iraqi News Agency.

According to officials who spoke to the Reuters news agency, fighting on Sunday killed at least 22 soldiers and 12 civilians in Ramadi and left an unknown number of militants dead. Video footage released by the Iraqi Defense Ministry showed late-night strikes on what it said were ISIS vehicles and hideouts.

“This is a fight that belongs to the Iraqis,” Kerry said toward the end of a visit to Jerusalem. “We are not, obviously, contemplating returning. We are not contemplating putting boots on the ground. This is their fight, but we’re going to help them in their fight.”

Kerry didn’t give details of what assistance the United States might provide but said it would do “everything that is possible.” After Maliki appealed in November for more U.S. support in fighting extremists, Washington sent 75 Hellfire missiles and promised to dispatch drones.

Fear and criticism

A local journalist in Fallujah said Sunday that the Iraqi army was shelling militant positions but that civilian areas in the city also had been hit.

“It is back to the same as it was in 2004,” said the journalist, referring to the major U.S. assaults. “Before 2004, there was only one cemetery in Fallujah. Afterwards, there were four cemeteries,” said the journalist, who spoke on the condition of anonymity for security reasons. “Now the people fear there will be eight cemeteries.”

Although Fallujah remained under the militants’ control Sunday, their grip on Ramadi appeared to be weakening.

An Iraqi military commander said it would take two to three days to expel militants from the two cities. Lt. Gen. Rasheed Fleih, who leads the Anbar Military Command, told state television that pro-government Sunni tribes are leading the operations while the army is offering aerial cover and logistics on the ground, the Associated Press reported.

The fall of Fallujah to the al-Qaeda-linked group prompted renewed criticism from U.S. lawmakers of the Obama administration, which had planned to leave several thousand American troops in Iraq after the 2011 pullout for training and counterterrorism operations. But the deal fell apart when Iraq’s parliament refused to guarantee the remaining U.S. military personnel immunity from prosecution in Iraqi courts.

“The thousands of brave Americans who fought, shed their blood and lost their friends to bring peace to Fallujah and Iraq are now left to wonder whether these sacrifices were in vain,” Republican Sens. John McCain (Ariz.) and Lindsey O. Graham (S.C.) said in a statement Saturday.

The Syria connection

ISIS’s tightened grip on Anbar gives it a swath of land straddling the Syrian border. “This is a fight that is bigger than just Iraq,” Kerry said Sunday. “The rise of these terrorists in the region, and particularly in Syria and through the fighting in Syria, is part of what is unleashing this instability in the rest of the region.”

But in Syria, the group is under pressure. In recent days, other rebel groups have joined forces to confront ISIS fighters, who include numerous non-

Syrians and have antagonized residents with their brutality.

On Sunday, Syrian rebels were locked in a third day of clashes with ISIS militants, during which the al-Qaeda-linked fighters have been expelled from a slew of towns and villages in the opposition-held north. ISIS pulled out of positions in the Syrian town of Adana, one of its strongholds, activists said. Meanwhile, near the Turkish border, fighting spread to the north-central province of Raqqah, an ISIS stronghold.

“ISIS appears to have sustained serious territorial losses,” said Charles Lister, a visiting fellow at the Brookings Doha Center in Qatar. “But much of this, for now, is likely a result of pragmatic withdrawals rather than outright military defeat.”

However, Lister warned, whatever takes place in the coming weeks, ISIS will probably remain in Syria in some form.

“Should it be entirely isolated by all other key fighting groups in Syria, its actions will likely become even more harsh than before,” he said.

The latest fighting occurred just weeks before scheduled Syrian peace negotiations. On Sunday, Kerry for the first time opened the door to Iranian participation in the talks.

Softening the U.S. hard line against any role for Iran if it refuses to endorse the ground rules for the Jan. 22 conference, Kerry suggested that Iran might be able to participate from the sidelines. Iran backs Syrian President Bashar al-Assad and has provided him with weapons and other military help.

“Could they contribute from the sidelines? Are there ways for them conceivably to weigh in?” Kerry asked rhetorically about the Iranians, who oppose the conference’s goal of establishing a transitional government in Syria.

Kerry suggested that Iran’s diplomatic office in Geneva might be able to serve as an unofficial participant.

The U.N. envoy organizing the conference, Lakhdar Brahimi, wants Iran included in the talks, and U.N. officials have suggested that Iran could play an unofficial role.

Although U.S. relations with Iran have improved since the successful early negotiations over the country’s nuclear program last year, the United States remains suspicious of Iran’s foreign-­policy goals in Syria and elsewhere.

Gearan reported from Jerusalem. Suzan Haidamous in Beirut contributed to this report.