The operation - Operation Kabda Bil Hadid or Operation Iron Fist - against "a known terrorist sanctuary" began early on Saturday in the town of al-Sada in the western province of al-Anbar, about 12km from the Iraq-Syrian border, the US military said in a statement.
A doctor in the main hospital in the border town of al-Qaim, Amir al-Obedi, said 10 people had been killed and 15 wounded since fighting began.
He said relatives of the wounded told him they had been attacked by US helicopters in al-Sada. The US military had no information about US or Iraqi casualties.
Residents were seen fleeing al-Qaim in cars, heading for the city of Ana, about 50km west, to escape fighting.
The US military said the operation is aimed at rooting out Iraqi al-Qaida fighters and disrupting their support systems in and around the town near the Syrian border.
In recent months, the statement said, fighters had established a base in al-Sada and used it to launch attacks against Iraqi civilians and US and Iraqi forces in the area.
The offensive was also aimed at stopping foreign fighters from entering the country from Syria, and at improving security in the area before Iraq's 15 October national referendum on the country's draft constitution, the military said.
Sunni imam killed
Meanwhile, police sources reported the imam of Baghdad's Arafat mosque, Salah Hassan Ayash, was shot dead at the wheel of his car in southeastern Baghdad, while an Iraqi army officer was also shot dead in his car in the district of Sadr City.
Car bombs continue to strike
Baghdad and surrounding cities
Three Iraqi policemen were killed and four wounded when a bomb targeted their patrol in the northern city of Kirkuk, police said.
"One vehicle was completely burnt with three policemen inside," said police Captain Mohammed Adil.
The second patrol car was badly damaged and its four occupants wounded, he added.
North of the capital in Baquba, a soldier and a bus driver were killed and three soldiers wounded when an unidentified armed group opened fire on their vehicle, a military source said.
A seven-year-old child was accidentally killed after police opened fire on the car he was travelling in suspecting it of being a car bomb, said the same source. Another passenger was wounded.
Two Iraqis, including a soldier, were killed in a bomb attack against an army patrol in Dujail, south of Baquba, said a local military source.
Two suspected fighters were also reported killed in Ishaqi, north of the capital, when the bomb they were planting appeared to have detonated prematurely, police said.
US, Danish soldiers killed
Elsewhere in Iraq, two US soldiers were killed in two separate bomb explosions in central Baghdad and near Baiji, 200km north of the capital, the US military said, without providing further details.
Also on Saturday, a Danish soldier was killed and two others were wounded when a roadside bomb blew up near a Danish military vehicle outside Basra in southern Iraq on Saturday, Iraqi police said.
Danish troops are stationed in
the southern cities of Iraq
An official at the Danish Ministry of Defence in Copenhagen confirmed that there had been an attack on a Danish military vehicle, but said he could not confirm the number of casualties.
"We are investigating further at this time," the military official said, asking that his name not be used.
Iraqi police officer Mohammed Uraibi told Reuters from the scene that the attack occurred just outside Basra, Iraq's second largest city.
If confirmed, the death would be the second among Danish troops serving in Iraq. Denmark has a contingent of about 500 soldiers serving in and around Basra, where British forces have overall command.
A Danish soldier was killed by friendly fire in August 2003.
The latest deaths brings to 1935 the number of US military personnel killed in Iraq since the US-led invasion of March 2003. Ninety-six British soldiers and another 102 military personnel from the multinational forces have been killed in the same period.
Meanwhile, the US military released about 500 Iraqi detainees from Abu Ghraib prison on the outskirts of Baghdad on Saturday, completing its plan to free a total of more than 1000 this week in honour of the Muslim holy month of Ramadan.
The releases began on Monday with the freeing of more than 500 detainees at Abu Ghraib prison, which gained international notoriety after a number of US military personnel were charged with humiliating and assaulting detainees at the facility.
The US released 500 Abu Ghraib
detainees for Ramadan
The Abu Ghraib actions, made at the request of Iraq's government, also appeared to be part of its effort to persuade Iraqis to vote in the referendum, especially the country's Sunni Arabs.
Many of them oppose the constitution, saying it would give Kurds living in the north and Shia in the south too much independence and control over Iraq's oil wealth, and leave Sunnis isolated in central and western Iraq.