And Cohen is expected to detail Trump's role in his hush-money payments to two women alleging during the 2016 election that they had been in sexual relationships with Trump.
The White House and Trump's Capitol Hill allies have been trying to discredit Cohen before he captures the attention of the US, casting him as a liar who cannot be trusted.
In a statement today, for example, White House press secretary Sarah Sanders called Cohen a "disgraced felon" and said "it's laughable that anyone would take a convicted liar like Cohen at his word."
Cohen pleaded guilty to several crimes last year, including lying to Congress about a Trump Organisation project in Moscow and orchestrating hush-money payments to women alleging the relationships. He has said he is finished with lying and is seeking to tell his story.
Reached by phone, Gaetz said the tweet "speaks for itself." He would not say whether Oversight Republicans intended to use this allegation during the hearing; Gaetz is not a member of the committee, but he is close to Republicans who are, including ranking member Jim Jordan, R.
Gaetz gave no proof or details about what he meant in his tweet. Asked whether his actions constituted witness intimidation, he balked: "Challenging the credibility and veracity of a witness is something that happens every day in America, and we need more of that in Congress when people intend to come and lie to us."
Lanny Davis, a lawyer for Cohen, said: "We will not respond to Mr Gaetz's despicable lies and personal smears, except to say we trust that his colleagues in the House, both Republicans and Democrats, will repudiate his words and his conduct.
"I also trust that his constituents will not appreciate that their congressman has set a new low - which in today's political culture is hard to imagine as possible," Davis added.
Criticism of Gaetz's comment lit up the Twittersphere.
Former congressman David Jolly, R, tweeted that Gaetz's message was "a new low in an age of low, and possibly a crime by a sitting Congressman."
Congressman David Cicilline, D, said on Twitter: "This isn't a scene from Godfather II. Witness intimidation is not going to work. The truth will come out."
Senator Christopher Coons, D, said on CNN that Gaetz's tweet constituted a lawmaker "badgering a witness . . . badgering of the type that seems more like it's out of a mob novel or a TV show."
Asked whether there should be consequences, he said he would leave it up to the House Oversight Committee. But Gaetz's behaviour, he continued, "lies well outside the boundaries of typical conduct here." If it were in a legal case, it would "arguably constitute witness intimidation," he said.
"I keep being shocked, and I keep thinking there is nothing that will shock me more . . . this tweet you've just put up on television is really something," Coons said.
Representative Bill Pascrell, D, called for the House Ethics Committee to investigate Gaetz's comments.
"One of my colleagues, a duly elected member of Congress, has taken to Twitter to intimidate a witness," he tweeted. "This is grossly unethical and probably illegal. House Ethics must investigate this disgrace and stain on our institution."
House Speaker Nancy Pelosi, D, stopped short of explicitly calling for an investigation into Gaetz's actions. But she gave him a stern warning.
"I encourage all members to be mindful that comments made on social media or in the press can adversely affect the ability of House committees to obtain the truthful and complete information necessary to fulfill their duties," she said.
"As a result, such statements can be construed as not reflecting credibility on the House, and the Committee on Ethics should vigilantly monitor these types of statements, which may not be protected by the speech or debate clause."
House Democrats have accused the President of trying to intimidate Cohen by suggesting in early January that Cohen's father-in-law had legal exposure as soon as he agreed to appear before the panel. Trump even called Cohen a "rat," mob-like language that Democrats say contributed to Cohen's concern for his and his family's safety.
Democrats say that's just proof that Trump, who is meeting with the leader of North Korea in Vietnam this week, is terrified of what secrets Cohen might reveal.
"Hard to combine disgusting and maybe criminal stupidity in one tweet but, hey, you did it," said Representative Sean Patrick Maloney, D. "Keep this up, Cohen's going to need a double cell."
Trump has said publicly that he is not concerned about Cohen's testimony, but privately his advisers said they are preparing for the President's former personal lawyer to delve into a number of issues that would be uncomfortable for Trump, including his hush-money payments to women.
White House officials are frustrated by the split-screen spectacle expected tomorrow, when Cohen's testimony is scheduled to begin at 4am NZT just after Trump finishes dinner in Hanoi with North Korean dictator Kim Jong Un, and just before the two leaders meet for a day of high-stakes nuclear arms negotiations.
"It's unfortunate this is happening now," said one White House official.
