Mark Warner of Virginia, the top Democrat on the Senate Intelligence Committee, said his panel or another committee would "absolutely" subpoena the tapes.
Later on Thursday, President Donald Trump went on to insist he is not under investigation, calling James Comey a "showboat" and a "grandstander", while calling the investigation a "charade".
The New York Times said Mr Comey told associates he declined to make a pledge of loyalty to Mr Trump when the president requested it while they dined just seven days after his inauguration.
It'd be unusual for the White House to elevate an FBI agent to the role of director, and McCabe during a Senate hearing last week broke with the White House's explanations for Comey's firing and its dismissive characterization of the Russian Federation investigation.
These reasons contradicted the White House's assertions - and even the widely-disseminated termination letter Trump sent Comey - that the dismissal was based on the recommendations of Attorney General Jeff Sessions and Deputy Attorney General Rod Rosenstein, who criticized Comey's handling of the email investigation into Hillary Clinton a year ago.
The president tweeted early Friday that "Again, the story that there was collusion between the Russians & Trump campaign was fabricated by Dems as an excuse for losing the election".
President Donald Trump threatened Friday morning to end White House press briefings, arguing that "it is not possible" for his staff to speak with "perfect accuracy" to the American public.
When he was asked, "Internally, from the president?" The lawmakers called for a new Federal Bureau of Investigation director without any political background and said Trump would need to hand over to Congress any taped conversations with Comey, if they exist. Clapper said America's founding fathers had created three co-equal branches of government with checks and balances, but with Trump as president, that was now "eroding". "I think the President wants loyalty to his country and to the rule of law", Spicer said. The latter apparently drafted a Memo on which the President based his decision to fire the FBI Director. But the nominee will require only a simple majority vote in the 100-member Senate, meaning Republicans can use their 52-48 majority to confirm the next director without needing Democratic votes. Sessions has since recused himself from Trump campaign-related investigative matters - though reports in the news media suggested he had a role in Comey's removal.
The next day, however, Trump said he had been considering firing Comey for a while, and that Rosenstein's memo had sealed the deal.
Fisher formerly served as assistant attorney general for the Justice Department's Criminal Division.
Clapper and Schumer made their comments on CNN's "State of the Union"; Graham spoke on NBC's "Meet the Press"; and Warner appeared on ABC's "This Week" and "Fox News Sunday".
At his Senate confirmation hearings, Attorney General Jeff Sessions, a staunch Trump supporter, failed to disclose campaign-trail meetings he had with a Russian ambassador.