But to many inside the White House, in other parts of government as well as to several close associates of the President, there is far less urgency or mystery. Nearly a dozen people from all of those walks of life told CNN they suspect it's merely Trump bluster and there's no such taping mechanism.
"It's all hot air," one person inside the White House said, speaking on condition of anonymity to describe internal water-cooler conversations. "Unless he had a recorder in his pocket, this didn't happen."
The President has played coy
. The White House press secretary declined again Monday to answer. Yet the final word won't come until Trump steps up and produces evidence or explains why none exists.
Never mind the political or legal consequences of making such a claim -- the people CNN interviewed all said they didn't assume such a recording device existed.
The Technical Services Division of the Secret Service is charged with sweeping the White House premises for listening devices. Stray audio equipment left behind by camera crews is known to draw suspicion, so it would be difficult for a formal recording system to go unnoticed.
'Just having a little fun'
A government official familiar with the operation told CNN that no recording device existed -- to their knowledge -- unless the President or one of his close aides used an iPhone or recorder on their own.
One person who has worked with Trump in the past declared there was "no chance" he had actually taped his conversations with Comey. He's "just having a little fun," this person said of Trump's tape antics.
CNN asked Sen. Lindsey Graham, who sits on the judiciary committee, if he personally thinks the White House tapes exist.
"I don't think so," he said, before taking a pause. "If there are, I'm on 'em!"
Yet others interviewed by CNN took it as the latest indication Trump is out of his depth in Washington.
"I don't think they exist," said one Republican on Capitol Hill, who -- like many other Republicans -- was perplexed by the President's decision to float the idea in the first place. "I think that was a miscalculation on his part."
Even if the tapes were real, the GOP aide added, they would probably prove Comey's side of the story and undercut the President's version of events.
"I'm pretty sure Comey wasn't lying here," the aide said.
Another Republican close to the White House said Trump's tape comments appeared similar to the hard-nosed tactics he took when his company was facing litigation.
"The tape thing is a total Trump real estate developer comment," the Republican said. "You just make it up, and you say that as a way of intimidating the other side."
The open question -- with potentially far-reaching consequences -- is whether Trump failed to understand the difference between playing the intimidation game in the boardroom versus the Oval Office.
"You're not in that playground anymore," another Republican close to the President said.
Nearly a month after firing off a suggestion on Twitter that such tapes may exist, Trump still refused to answer the question directly. But even he appears to be hinting that it's all a charade.
"I'll tell you about it over a very short period of time," Trump said in the Rose Garden on Friday. "You're going to be very disappointed when you hear the answer, don't worry."
If Trump's initial tapes tweet was meant as an intimidation tactic, it almost certainly backfired.
That tweet led the former FBI director to leak portions of the memos he had kept to document his conversations with Trump -- in the hopes of prompting the appointment of a special counsel.
"The President tweeted on Friday, after I got fired, that I better hope there's not tapes," Comey said in his testimony before the Senate intelligence committee. "I woke up in the middle of the night on Monday night, because it didn't dawn on me originally that there might be corroboration for our conversation. There might be a tape."
On May 17, five days after Trump tweeted about the tapes, the Justice Department named Robert Mueller as special counsel to oversee the Russia investigation.
And last Friday, the day after Comey testified on Capitol Hill and the same day Trump again refused to say whether tapes existed -- the two leaders of the Russia investigation in the House fired off a request to the White House: If there are recordings, Congress wants to see them by June 23.