WASHINGTON — The Trump administration is poised to impose steel and aluminum tariffs on the European Union, and possibly on Canada and Mexico, this week when a temporary exemption expires as trade talks remain at an impasse, according to a person familiar with the White House discussions.

The threat of tariffs was supposed to force trading partners into quickly agreeing to United States concessions, including limiting imports of steel and aluminum and other trade terms that favor American companies and workers. In addition to the metal tariffs, President Trump has threatened to impose levies on imported automobiles, the bulk of which comes from allies like Canada, Mexico, South Korea, Japan and the European Union.

The White House has tried to leverage the tariffs to get Canada and Mexico to agree to its demands on a rewrite of the North American Free Trade Agreement, and it has pressured Europe to make trade concessions on other fronts, including cutting the amount of metals the bloc exports to the United States.

So far, the Trump administration’s aggressive approach appears to have only hardened the resolve among the United States’ closest allies to fight back against Mr. Trump’s demands.