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 move comes after President Trump tweeted what appeared to be a thinly veiled threat to the former FBI chief.
Donald Trump warned James 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 existence of any tapes.
Charles Schumer also warned that Senate Democrats might refuse to vote on a new FBI director until a special prosecutor is named to investigate alleged Russian meddling in the US election.
The FBI is investigating the allegations and possible ties between Russia and the Trump campaign.
Donald Trump denies any such links and says James Comey had assured him he was not being investigated. The president says he fired James Comey because he was not doing a good job.
Democrats, however, have accused President Trump of firing James Comey to try to thwart the FBI inquiry.
Charles Schumer told CNN that if any tapes existed “the president should turn them over immediately. To destroy them would be a violation of law”.
“If there are no tapes, he should apologize to both Jim Comey and the American people for misleading them,” he added.
Meanwhile, Senator Graham told NBC that Donald Trump’s tweet was “inappropriate” and called on the president to “back off and let the investigation go forward”.
“You can’t be cute about tapes,” he said.
“If there are any tapes of this conversation, they need to be turned over.”
President Trump has said he could announce a replacement for James Comey later this week.
Eleven people are reportedly being considered for the position, which requires confirmation in the Senate, and interviews began on May 13.
Those under consideration include acting FBI Director Andrew McCabe, New York Appeals Court Judge Michael Garcia, Republican Senator John Cornyn and senior lawyer Alice Fisher.
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