The trip also seemed to represent another step, after a meeting between Mr. Trump and Mr. Abadi in Washington several weeks ago, to assure the Iraqis that the United States is a reliable ally, given that the relationship between the Trump administration and Iraq began on rocky footing.

First, in a speech at the C.I.A. shortly after his inauguration in January, Mr. Trump expressed regret that the United States had not taken Iraq’s oil after the 2003 invasion, and he suggested there still might be a chance to do so. Then, he placed Iraq on a list of seven Muslim-dominated countries whose citizens were barred from traveling to the United States.

That ban was blocked by the courts and later replaced with a new one that did not include Iraq on the ban list. But Iraqis still view the initial listing as an insult at a time when the two countries are working together to fight the Islamic State in the battle for Mosul.

Iraq may yet take on greater importance for the Trump administration than just its part in the fight against the Islamic State. With administration officials having signaled a desire to take a tougher stance on Iran, which like Iraq is majority Shiite and whose power inside Iraq is paramount, Iraqi officials have worried that their country could again become a place for the two foreign powers to settle their differences.

After the American invasion of 2003, Iraqi Shiite militias aligned with Iran killed hundreds of Americans, and those same militias have taken on an important role in fighting the Islamic State. They are now essentially a parallel security structure within the state, even though they are directed by Iran.

There are already signs that the Trump administration sees Iraq as a place to push back on the growing power of Iran across the region, where Tehran has expanded its power through proxies in four countries: Iraq, Syria, Lebanon and Yemen. In late February, for the first time in years, the foreign minister for Saudi Arabia, a Sunni power and Iran’s chief regional rival, visited Baghdad in a trip encouraged by American officials.

That was seen by Iraqi officials as a signal by the Americans that they are intent on nudging the Saudis to take a stronger role in Iraq, whose Sunni minority has been alienated from the power centers of Baghdad. But Iraqi officials have said they worry that that will only exacerbate sectarian tensions in Iraq, and that the last thing the country needs is more foreign meddling.