But an Iraqi doctor in the region told Aljazeera that the number of casualties, which he said included women and children, was not immediately known.
Also on Sunday, several mortar shells exploded near the Interior Ministry in Baghdad, wounding four civilians, said police Colonel Hussein al-Janabi.
About 1000 US troops, backed by attack helicopters and warplanes, began the offensive in the western village of al-Sada on Saturday morning, also hoping to close supply routes to fighters and stem violence ahead of Iraq's crucial vote on a new constitution this month.
On Sunday morning, US forces appeared to be widening the offensive to two nearby towns, residents said in telephone interviews.
In Karabila, troops using loudspeakers were warning residents to stay inside for their own safety, and in Rumana, a town on the other side of the Euphrates River, helicopters fired on a few houses, sending plumes of black smoke up into the air, the witnesses said, speaking on condition of anonymity out of concern for their security.
A doctor in the region said the number of casualties in the US offensive was not immediately known.
"Some people, mostly women and children, have been killed and injured in these areas. We don't know the exact number of casualties," Dr Ammar al-Marsumi, an Iraqi doctor at al-Qaim hospital, told Aljazeera.
"We cannot transfer casualties to al-Qaim hospital as roads are closed and bridges are destroyed," he added.
The US military attacks were close
to al-Qaim in western Iraq
Al-Marsumi said there were no clashes or military presence in the centre of al-Qaim city.
"US strikes only hit the three villages near al-Qaim," he added.
On Saturday, US aircraft firing missiles struck houses and cars, as the forces moved into al-Sada, according to residents.
US military reports
The first day of the US military offensive Operation Iron Fist saw clashes with fighters in and around al-Sada, 12km east of the Syrian border, the military said on Sunday in its first full report on the fighting.
Fighters drove two vehicles towards one US marine position, dismounted and began to attack with small-arms fire, the military said. One of the vehicles was found to be rigged with explosives. The gun battle left four fighters dead, the military said. A fifth surrendered.
Marines found and destroyed a roadside bomb on the outskirts of al-Sada, which is on the banks of the Euphrates River, and a US tank destroyed a vehicle carrying a bomb southwest of the village, the military said.
North of al-Sada, US forces killed three members of al-Qaida in Iraq after they attacked a US checkpoint with small-arms fire, the military said.
Another fighter was killed when a US Cobra helicopter destroyed a vehicle after its driver fired on a marine position with a rocket-propelled grenade, the military said.
A second vehicle believed to be carrying grenades was destroyed by missile fire from the Cobra, but its driver and passenger escaped, the military said.
No US casualties were reported.
US soldiers captured?
A website statement reportedly attributed to the al-Qaida group in Iraq, however, claimed that two US marines had been captured by its forces.
US marines have continued their
offensive in western Iraq
"Al-Qaida soldiers succeeded in kidnapping two US marines ... and is giving the infidels 24 hours to release female Sunni Muslim prisoners ... or they (Western forces) should not bother to look for their children," the statement said on a Web site generally used by the group.
The authenticity of the website statement has not yet been verified.
A spokesman for US forces in Iraq, Lieutenant Colonel Steve Boylan, said: "I have not heard anything about any of our folks being taken. I would suspect that these are unfounded rumours, as that is what has happened in the past."
Fourth US offensive
Operation Iron Fist is the fourth large US offensive in the border area since May.
In al-Sada, 290km northwest of Baghdad, US snipers took positions on rooftops, as troops went house to house on Saturday blasting open doors, in a hunt for fighters, a correspondent for CNN embedded with the troops reported.
On Sunday, the correspondent said many residents appeared to have fled the village days earlier, having heard that a US offensive was likely to occur.
Al-Sada is an isolated village of about 2000 people, with one main road and nearly 200 houses scattered in a rural area near the town of al-Qaim in Iraq's western province of al-Anbar.