On Fox News, Senator Mike Lee, Republican of Utah, said if there are recordings, it was “probably inevitable” that they would be turned over to Congress, which is investigating possible connections between Mr. Trump’s campaign and Russia.

Both senators, along with others, were responding to the president’s warning to Mr. Comey, in a Twitter post on Friday, that he had “better hope that there are no ‘tapes’ of our conversations before he starts leaking to the press!”

The minority leader, Senator Chuck Schumer of New York, said that President Trump should turn over such tapes “immediately,” if they exist, while Senator Mark Warner of Virginia, the ranking Democrat on the committee investigating the Russia ties, said that such recordings must be preserved.

Mr. Schumer also used the controversy to repeat his call for a special prosecutor to lead the Russia investigation, and left open the possibility that Democrats might withdraw support for a new F.B.I. director if the Justice Department did not appoint an independent prosecutor.