There is widespread agreement among Shiite political blocs, with whom Mr. Sadr would have to ally to form a government, to continue a program backed by international troops to train and equip Iraqi security forces. Trainers include American, Italian and Spanish advisers, with equipment paid for by the United States. And having NATO serve as the public representative for the American-led mission in Iraq could serve as a workaround for Mr. Sadr’s sensitivities, officials say.

Should the Trump administration and a government loyal to Mr. Sadr align, it would not be the first time American troops have had to develop a working relationship with Iraqis who were once considered the enemy. The partnering of United States forces with Sunni insurgents known as Sahwa, or the Awakening, against Al Qaeda in Iraq was a defining turning point in the war more than a decade ago.

But the Pentagon will have its own balancing act to perform back in Washington. Mr. Trump has already expressed his desire to bring American troops home soon from Syria; officials said the president has given the Defense Department six months to wrap up its mission there. Military officials had hoped that an American troop presence in Iraq could keep in contact with allied forces across the border in Syria.