Senate Democratic leader Charles Schumer warned that destroying any tapes would break the law.
Republican Senator Lindsey Graham said the White House needed to "clear the air" about whether tapes existed.
The comments come after Mr Trump tweeted what appeared to be a thinly veiled threat to the former FBI chief.
He warned Mr Comey last week against talking to the media, saying he had "better hope there are no tapes" of their conversations.
The White House has neither confirmed nor denied the presence of any tapes.