WASHINGTON—The U.S. has intercepted an order from Iran to militants in Iraq to attack the U.S. Embassy and other American interests in Baghdad in the event of a strike on Syria, officials said, amid an expanding array of reprisal threats across the region.

Military officials have been trying to predict the range of possible responses from Syria, Iran and their allies. U.S. officials said they are on alert for Iran's fleet of small, fast boats in the Persian Gulf, where American warships are positioned. U.S. officials also...