And he is fixated on an issue that he raged about both during the campaign and in the weeks after his victory: the integrity of the country’s voting procedures. After his loss by about three million votes in the popular vote, and the questions about his legitimacy that have followed, he is now pushing for an investigation that could cost millions of dollars.

On Monday, Mr. Trump shocked congressional leaders at a White House reception by recounting a story he said he had heard from Bernhard Langer, a professional golfer, about voters in Florida he assumed to be illegal. (Mr. Langer later said that he had never talked to Mr. Trump about voting, but had only passed along an anecdote he had heard from a friend to another friend, “who shared it with a person with ties to the White House.”)

On Wednesday, the president explained on prime-time television why he had lost the popular vote. “Of those votes cast, none of them come to me,” Mr. Trump told ABC News about what he has called millions of illegal votes. “They would all be for the other side.”

And then, early Friday, he saw a segment on CNN’s early-morning show in which Mr. Phillips argued without evidence that as many as three million people had voted illegally. “Look forward to seeing final results of VoteStand,” Mr. Trump wrote on Twitter 47 minutes after the CNN interview, referring to Mr. Phillips’s app. “Gregg Phillips and crew say at least 3,000,000 votes were illegal. We must do better!”

Election experts aligned with both major parties say that no such mass election fraud took place.

“We are not aware of any evidence that supports the voter fraud claims made by President Trump, but we are open to learning more about the administration’s concerns,” the National Association of Secretaries of State said in a statement.