The Trump administration has intervened in recent weeks in two Middle East disputes that have hobbled regional oil production, aiming to boost global supplies as Iranian oil sanctions take effect, according to officials familiar with the efforts.

Washington is trying to broker a deal between Saudi Arabia and Kuwait over a contested strip of oil-rich land called the “neutral zone” between the two countries, these officials said. U.S. officials are also trying to persuade the federal government of Iraq to allow exports of oil...