Yesterday, 42 days later, he finally did. As most in Washington had anticipated, Trump said he did not have any such tapes.
The incident highlights a new reality for Washington, which now must spring into action to bolster or refute presidential assertions of dubious origin and with no evidence to back them up. In many cases, the claims have also had the opposite effect than what the President likely intended - feeding into doubts about his credibility, deepening his legal woes and generating unflattering accounts that dominate the news for weeks at a time.
And even when Trump has walked back a questionable comment, he has sometimes planted a new and similarly unsubstantiated claim. In denying yesterday that he had created "tapes" of his conversations with Comey, for example, Trump also suggested that he may have been surveilled.
"With all the recently reported electronic surveillance, intercepts, unmasking and illegal leaking of information, I have no idea whether there are 'tapes' or recordings of my conversations with James Comey," Trump wrote in one tweet, before denying that he had created any.
Before the tapes, there was Trump's unfounded claim that former President Barack Obama "wiretapped" him in Trump Tower during the campaign - setting off a flurry of official inquiries from Congress. His oft-repeated campaign promise that the wall along the southern US border would be paid for by Mexico is one that lawmakers in Trump's own party believe will never happen - yet they and others in the Government continue to look for some way to help the President save face.
Trump has also repeatedly claimed that millions of illegal immigrants voted in the last election, with no proof. Yet in an effort to validate his comments, the Trump Administration has created a commission aimed at investigating his claim of widespread voter fraud.
"What happens with the President is he shoots himself in the foot and soon the gangrene spreads to the entire body politic," said Norm Eisen, a former US ambassador to the Czech Republic and an ethics czar in the Obama Administration. "This is going to be the new normal: elements of the President's own executive branch openly, or indirectly through leaks, responding to these false tweets."
After Trump raised the prospect of Comey-related tapes, exasperated lawmakers in both parties pledged to find out one way or another. "I don't have the foggiest idea," Democratic Senator Mark Warner said on ABC News the following Sunday morning.
But the most significant consequences were yet to come.
Comey told lawmakers in testimony this month that as he laid awake in his northern Virginia bed a week after he was summarily fired, he decided to act - in large part because of Trump's tweet.
"It didn't dawn on me originally that there might be corroboration for our conversation. There might be a tape," he said, explaining why he leaked memos of his conversations with Trump to the media. He also testified, "Lordy, I hope there are tapes!"
Comey's memos prompted the appointment of special counsel Robert Mueller, a former FBI director, to investigate possible collusion between Trump campaign associates and Russians who interfered in the election. The Washington Post has also reported that Mueller is investigating whether Trump attempted to obstruct the investigation.
"There's nothing criminal or illegal about bluffing," said Alan Dershowitz, a Harvard Law School professor who has often defended Trump against various allegations. "I don't think he would have said he had tapes if he had them."
But Dershowitz acknowledged that the tweet may have been a short-sighted attempt to ensure that Comey was careful about his public statements on Trump.
"I don't know whether it was an unforced error or a tactic, but it could have been both: a tactic that turned out to be an unforced error," Dershowitz said.
"He should have thought through all of that. I very often keep contemporaneous memos, particularly when I'm dealing with people who have credibility issues.
"Lawyers do that," he added.
A similar dynamic played out in March, when Trump blasted out another shocker of a tweet claiming that Obama had wiretapped him - animplausible assertion that government officials and lawmakers moved quickly to deny.
But among Trump loyalists in the White House and in Congress, there was a spirited effort to validate the claim.
Three White House officials unearthed classified documents that suggested the Obama Administration officials may have "unmasked" the names of Trump campaign associates that were contained in classified intelligence reports.
Intelligence experts note that unmasking is a legal practice if done properly and completely different from Trump's claim that he was illegally "wire tapped". But armed with the documents procured by the White House, Republican Devin Nunes, chairman of the House Intelligence Committee and a member of Trump's transition team, set out to defend Trump's tweets.
Nunes later told Fox News host Sean Hannity that he felt obligated to brief the President on the unmasking issue because he was "taking a lot of heat in the news media" for his wiretapping tweets.
To intelligence experts, the controversy was an attempt by Trump loyalists to confuse two entirely separate issues - illegal surveillance and legal "unmasking" of the names of American individuals - in order to defend the President.
"The notion that President Obama could instruct the intel community to set up a tap on Mr Trump's offices is preposterous on its face. He doesn't have that authority," said Robert Deitz, a former general counsel at the National Security Agency and the Defence Department. "One of the things that's interesting about Washington is that it's a little bit of Alice in Wonderland: You hear something or you see something in the press and you try to make sense of it."
The Trump Administration has moved to accommodate the President's dubious rhetoric in other ways.
Trump has repeatedly insisted without evidence that he lost the popular vote because of millions of illegal immigrant voters. That in turn led the White House to create a commission to study the issue - an effort widely dismissed as a sham but which nevertheless is slated to produce a report of its findings next year.
A similar phenomenon has taken hold with Trump's proposed border wall. The President, lawmakers and his aides have floated a number of schemes to make his campaign promise come true, including initially financing the wall with solar panels or a border adjustment tax plan. But with Mexico refusing to even entertain the idea of funding - and with Senate Majority Leader Mitch McConnell answering "no", when asked if Mexico would be paying up - Trump hasn't dropped the issue.
