Falluja is one of the main cities in Anbar province, which was the centre of the Sunni-led fight against US forces in the months following America’s March 2003 invasion.
Also on Thursday, two US soldiers were killed and nine Iraqi’s wounded in a suicide car bombing in the northern city of Mosul.
Elsewhere in northern Iraq, four fighters were killed and 32 detained in US operations targeting al-Qaeda in the north of the country, the US military said.
The largest raid occurred in the disputed city of Kirkuk.
US ground forces came under fire while surrounding a building where a targeted al-Qaeda courier was believed to be hiding and killed those firing at them, the military said.
They also said that 16 other people suspected of links to al-Qaeda were detained in an earlier raid.