Mr. Trump, who has already fired his national security adviser and acting attorney general, could dismiss Mr. Comey, but that would probably lead to significant backlash from lawmakers and federal authorities who would see such a move as an attempt to influence the Russia investigation.

Mr. Trump started the controversy early Saturday morning with a series of Twitter posts.

“Terrible! Just found out that Obama had my ‘wires tapped’ in Trump Tower just before the victory,” Mr. Trump said. “Nothing found. This is McCarthyism!”

Mr. Trump added: “This is Nixon/Watergate. Bad (or sick) guy!”

The White House has not officially said what led Mr. Trump to make the claims. But administration officials have acknowledged that they were primarily prompted by unverified claims by Breitbart News and conservative talk radio hosts that secret warrants were issued authorizing tapping the phones of Mr. Trump and his aides.

Mr. Comey was said to be disturbed by Mr. Trump’s claims about Mr. Obama, which insinuated that the F.B.I. had broken the law and raised the public’s expectations about how much evidence federal authorities might have had on Mr. Trump. For the Justice Department to have obtained a warrant to eavesdrop on him, federal authorities would have had to prove to a judge that there was significant evidence that he was breaking the law or was the agent of a foreign power.

Along with asking the House and Senate Intelligence Committees to investigate whether Mr. Obama eavesdropped on Mr. Trump, Mr. Spicer called on the committees to investigate what he called a steady stream of national security leaks since Mr. Trump took office. Mr. Spicer said the White House would not commit to accepting the findings of those investigations.

“I don’t think you would ever just blanketly say, ‘I’m going to accept any outcome,’” Mr. Spicer said.

Some Republicans said on Monday that spying was a hallmark of Mr. Obama’s administration, claiming that during his time in office the Internal Revenue Service targeted conservative groups. Other Republicans defended the impartiality of the Justice Department and F.B.I.