The U.S. Supreme Court allowed Donald Trump on Thursday to keep two Democratic members of federal labor boards away from their posts while their challenge to his firing of them proceeds in a legal dispute that tests the Republican president's power over independent government agencies.

Mr. Trump had pressed hard for the bill’s passage, visiting the Capitol on Tuesday to pitch Republicans on the legislation, and meeting with holdouts on Wednesday at the White House. Earlier on Wednesday, his administration put out a statement calling the failure to pass the bill “the ultimate betrayal.”