The White House showed no indication that it would back down from Mr. Trump’s claims. On Sunday, the president demanded a congressional inquiry into whether Mr. Obama had 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 Congress should examine them as part of its investigations into Russia’s meddling in the election.

In addition to being concerned about potential attacks on the bureau’s credibility, senior F.B.I. officials are said to be worried that the notion of a court-approved wiretap will raise the public’s expectations that the federal authorities have significant evidence implicating the Trump campaign in colluding with Russia’s efforts to disrupt the presidential election.

Mr. Comey has not been dealing directly with Attorney General Jeff Sessions on the matter, as Mr. Sessions announced on Thursday that he would recuse himself from any investigation of Russia’s efforts to influence the election. It had been revealed on Wednesday that Mr. Sessions had misled Congress about his meetings with the Russian ambassador during the campaign.

Mr. Comey’s behind-the-scenes maneuvering is certain to invite contrasts to his actions last year, when he spoke publicly about the Hillary Clinton email case and disregarded Justice Department entreaties not to.

It is not clear why Mr. Comey did not issue a statement himself. He is the most senior law enforcement official who was kept on the job as the Obama administration gave way to the Trump administration. And while the Justice Department applies for intelligence-gathering warrants, the F.B.I. keeps its own records and is in a position to know whether Mr. Trump’s claims are true. While intelligence officials do not normally discuss the existence or nonexistence of surveillance warrants, no law prevents Mr. Comey from issuing the statement.