The US commander of the joint force, Colonel Stephen Davis, said late on Sunday that his troops had moved about halfway through Husaybah, a market town along the Euphrates river about 320km northwest of Baghdad.
At least 36 fighters have been killed since the assault began on Saturday and about 200 men have been detained, Davis said.
He did not give a breakdown of nationalities of the detainees.
The New York Times, which has a journalist embedded with the American forces, reported on Monday that one US marine was killed and three were wounded the day before.
Firsts US fatality
In Baghdad, the US military said it could not immediately confirm that, but it often delays the release of casualty information until relatives of the victims are notified.
CNN, which also has a reporter accompanying the offensive, said that at least one Iraqi soldier has been wounded and that as many as 80 fighters have died in the fighting.
In a live report from the scene on Monday morning, CNN said the house-to-house battles were continuing, with ground forces supported by Humvees and tanks working their way through the narrow streets of the bleak desert town and with US jets and helicopters flying overhead.
US officials have described Husaybah, which used to have a population of about 30,000, as a stronghold of al-Qaida in Iraq, led by Jordanian Abu Musab al-Zarqawi.
Scores of Iraqis fled the town to
Husaybah had long been identified as an entry point for foreign fighters, weapons and ammunition entering from Syria.
Several people identified as key al-Qaida in Iraq officials have been killed in recent air strikes in the Husaybah area, the US military said.
Most were described as “facilitators” who helped smuggle would-be bombers from Syria.
Falih Abd al-Karim, an Iraqi journalist, told Aljazeera that US and Iraqi forces were positioned in al-Sikak neighbourhood and north and south of 12 Rabia al-Awal neighbourhood in central Husaybah.
This came after US warplanes on Sunday evening targeted al-Jamahir, al-Risala and other neighbourhoods in the town, destroying houses, and killing and injuring dozens of people, he said.
People were told to leave on foot
The bodies remained under the debris of the houses because US forces do not allow burials or transfer of the injured to hospitals, Abd al-Karim added.
The US shelling has demolished government buildings, including al-Jamahir primary school, al-Qaim preparatory school for boys, the educational supervision building, al-Qaim post office and communication centre, al-Qaim education directorate and two mosques in the city, he said.
Snipers have taken positions on tall buildings, forcing residents to stay inside, Abd al-Karim said.
Al-Qaim residents have no food, medicine, fuel, water or electricity, he added.
Al-Qaida leaders killed
For its part, the US military announced on Monday that it had killed two regional al-Qaida leaders in Iraq operating in the Husaybah area during air raids on 31 October near the towns of Karabilah and Obaidi.
It identified one of them as Abu Umar, who helped smuggle foreign fighters into the region and stage deadly roadside bomb attacks against Iraqi and American forces.
The other was Abu Hamza, who commanded several al-Qaida cells and helped launch attacks against international forces, including ones based at US Camp Gannon in the Husaybah area, the military said.