The legitimacy of President Trump’s tariffs is being questioned by U.S. courts, but the president is showing no signs of backing off his favorite tool.

On Wednesday, the tariffs that Mr. Trump imposed on foreign steel and aluminum are set to double to 50 percent, a move that the president has said will better protect domestic metal makers.

The increase comes as the U.S. government this week faces off with states and businesses that have sued over the president’s tariffs. Both sides will be required to submit more information as judges work toward final decisions on the legality of Mr. Trump’s steepest tariffs.

The U.S. Court of International Trade ruled last week that some of the steep tariffs that Mr. Trump had imposed were illegal, a significant setback for the president’s agenda. But a separate court temporarily paused that decision less than 24 hours later. As judges weigh that appeal, the tariffs in question — which include the levies Mr. Trump imposed on Canada, Mexico and China for what he said was their role in the fentanyl trade, as well as the global tariffs Mr. Trump announced, and then quickly paused, in April — are expected to remain in effect until at least June 9.