President Trump, a day after leveling a widely disputed allegation that President Barack Obama had ordered the tapping of his phones, on Sunday demanded a congressional inquiry into whether Mr. Obama abused the power of federal law enforcement agencies before the 2016 presidential election.

In a statement from his spokesman, Mr. Trump called “reports” about the wiretapping “very troubling” and said that Congress should examine them as part of its investigations into Russia’s meddling in the election.

“President Donald J. Trump is requesting that as part of their investigation into Russian activity, the congressional intelligence committees exercise their oversight authority to determine whether executive branch investigative powers were abused in 2016,” Sean Spicer, the White House press secretary, said in the statement.

Mr. Spicer, who repeated the entire statement in a series of Twitter messages, added that “neither the White House nor the president will comment further until such oversight is conducted.”

A spokesman for Mr. Obama and his former aides have called the accusation by Mr. Trump completely false, saying that Mr. Obama never ordered any wiretapping of a United States citizen.

“A cardinal rule of the Obama administration was that no White House official ever interfered with any independent investigation led by the Department of Justice,” Kevin Lewis, Mr. Obama’s spokesman, said in a statement on Saturday.

Mr. Trump’s demand for a congressional investigation appears to be based, at least in part, on unproved claims by Breitbart News and conservative talk radio hosts that secret warrants were issued authorizing the tapping of the phones of Mr. Trump and his aides at Trump Tower in New York.

In a series of Twitter messages on Saturday, the president seemed to be convinced that those claims were true. In one post, Mr. Trump said, “I’d bet a good lawyer could make a great case out of the fact that President Obama was tapping my phones in October, just prior to Election!”

On Sunday, Sarah Huckabee Sanders, the deputy White House press secretary, said the president was determined to find out what had really happened, calling it potentially the “greatest abuse of power” that the country has ever seen.

“Look, I think he’s going off of information that he’s seen that has led him to believe that this is a very real potential,” Ms. Sanders said on ABC’s “This Week” program. “And if it is, this is the greatest overreach and the greatest abuse of power that I think we have ever seen and a huge attack on democracy itself. And the American people have a right to know if this took place.”

Senior law enforcement and intelligence officials who worked in the Obama administration have said there were no such secret intelligence warrants regarding Mr. Trump. Asked whether such a warrant existed, James R. Clapper Jr., a former director of national intelligence, said on NBC’s “Meet the Press” program, “Not to my knowledge, no.”

“There was no such wiretap activity mounted against the president-elect at the time as a candidate or against his campaign,” Mr. Clapper added.

Mr. Trump’s demands for a congressional investigation were initially met with skepticism by lawmakers, including Republicans. Appearing on CNN’s “State of the Union” on Sunday, Senator Marco Rubio, Republican of Florida, said he was “not sure what it is that he is talking about.”

“I’m not sure what the genesis of that statement was,” Mr. Rubio said.

Pressed to elaborate on “Meet the Press,” Mr. Rubio said, “I’m not going to be a part of a witch hunt, but I’m also not going to be a part of a cover-up.”