WASHINGTON — The American military downed an Iranian drone in the Strait of Hormuz on Thursday in what President Trump called an act of self-defense, just hours after Iran’s chief diplomat offered a modest road map for easing tensions with the United States.

Officials said the uncrewed, relatively small drone came within 1,000 yards of the Boxer, a United States amphibious assault ship in the strait. It was not known if the drone was armed, but a Pentagon spokesman, Jonathan Hoffman, said that it had “closed within a threatening range” before being shot down over international waters.

Mr. Trump described it as “the latest of many provocative and hostile actions by Iran.”

Iran’s foreign minister, Mohammad Javad Zarif, appeared to brush off the president’s broadside. “The drone issue is being investigated, but based on the latest news I have from Tehran, we have no information about losing a drone,” he told reporters at the United Nations.

At the United Nations headquarters, he said he was willing to discuss possible ways out of the crisis that erupted after Mr. Trump last year withdrew the United States from a 2015 nuclear deal between Iran and world powers.