Those efforts failed due to pushback from his own appointees in the Justice Department, who refused to file what they viewed as a legally baseless lawsuit in the Supreme Court. Later, other senior department officials threatened to resign en masse should Mr. Trump fire then-acting Attorney General Jeffrey Rosen, according to several people familiar with the discussions.

Senior department officials, including Mr. Rosen, former Attorney General William Barr and former acting Solicitor General Jeffrey Wall refused to file the Supreme Court case, concluding that there was no basis to challenge the election outcome and that the federal government had no legal interest in whether Mr. Trump or Mr. Biden won the presidency, some of these people said. White House counsel Pat Cipollone and his deputy, Patrick Philbin, also opposed Mr. Trump’s idea, which was promoted by his outside attorneys, these people said.

“He wanted us, the United States, to sue one or more of the states directly in the Supreme Court,” a former administration official said. “The pressure got really intense” after a lawsuit Texas filed in the Supreme Court against four states Mr. Biden won was dismissed on Dec. 11, the official said. An outside lawyer working for Mr. Trump drafted a brief the then-president wanted the Justice Department to file, people familiar with the matter said, but officials refused.