WASHINGTON—President Donald Trump plans to apply his steel and aluminum tariffs globally and won’t exempt allies such as Canada and Europe, a senior White House official said Friday, an approach that is likely to intensify protests over the move.

The statement that there would be no exceptions to the duties came as Trump aides started to flesh out the president’s broad Thursday announcement, in which he said the U.S. plans to impose tariffs of 25% on steel and 10% on aluminum. Officials are scrambling to finalize many details...