"We move around a lot. We've been all over the country," one of the US servicemen told Reuters on the bridge, about 45 km (28 miles) southeast of Mosul. He said the Iraqis were making quick progress in repairing the span, and that the American troops would leave the area within days.
Loath to become mired in another conflict overseas, the White House has insisted there will be no American "boots on the ground" in Iraq, but current troop levels are approaching 5,000.
That is still a fraction of the 170,000 deployed at the height of the nine-year occupation that toppled Saddam Hussein in 2003, sparking an al Qaeda-backed insurgency and throwing the country into a sectarian civil war.
President Barack Obama withdrew US troops from Iraq but they returned in 2014 after the Iraqi army fled Islamic State's advance through a third of the country despite billions of dollars in US aid and training.
The United States is conducting an extensive air campaign over Iraq and also covert special forces raids against the jihadists behind their frontlines.
But Washington says the focus of its troops in the country is to train, advise and equip local forces - Iraqi military and police, Kurdish Peshmerga and Sunni tribal militias, which are both battling Islamic State - and that US servicemen there have no combat role.
Advisers from the United States and other countries from an international coalition fighting Islamic State were initially confined to a few military bases across Iraq, but as the campaign progressed and Mosul comes into focus, Americans have inched closer to the action.