But Cohen attorney Lanny Davis called the attack baseless, and ABC News' chief legal analyst also said Trump's lawyers may have difficulty backing up their claim.
Trump then approved a meeting taking place, CNN reported on Thursday night, citing sources with knowledge of the matter.
Cohen quickly became embroiled in questions about Trump's ties to Russia and Russian interference in the 2016 campaign. If he mentioned the crucial detail to House investigators it was not included in their massive report.
Calling it a "dis-barrable offense", Giuliani told CBS' "Face The Nation" that "I did not know that he taped conversations surreptitiously".
Noting that Cohen himself is under federal investigation, Trump tweeted at one point: "Sounds to me like someone is trying to make up stories in order to get himself out of an unrelated jam (Taxi cabs maybe?)".
The criticism Giuliani is now leveling at Cohen follows a series of reports that have led the President to lash out at his former lawyer.
Cohen is facing federal charges related to his business practices in New York City, but a prolonged dispute over evidence in the case means he has not yet been presented with a formal list of charges.
Federal prosecutors in NY are investigating Cohen for possible bank and tax fraud, and for possible campaign law violations linked to a $130,000 payment to adult film star Stormy Daniels, who has claimed she had a sexual encounter with Trump, and other matters related to Trump's campaign, a person familiar with the investigation has told Reuters.
He later added, "I don't see how he has any credibility".
But after the release of the Trump-Cohen recording, Trump criticized his former attorney, saying on Wednesday, "What kind of a lawyer would tape a client?"
"Donald Trump Jr. has been professional and responsible throughout the Mueller and Congressional investigations", Trump Jr.'s attorney told the network.
Giuliani, who called Cohen "an honest, honorable lawyer" as recently as May, has made a sport out of bashing Cohen in recent days.
The June 2016 meeting was arranged through a publicist, who told Trump Jr in emails that a senior Russian official "offered to provide the Trump campaign" with damaging information about Ms Clinton. There have been earlier suggestions, from elsewhere, that Trump knew about the meeting. There was only one reason he agreed to a breakfast meeting, Catsimatidis said: because Cohen name-dropped Trump.
That would create another headache for Trump's defense team.
"I completely understand why under ordinary circumstances prosecutors would not approve of what I did", Davis said in a separate interview Wednesday.
The rift in talks between Trump's legal team and the Mueller team comes amid recent efforts by Trump to personally discredit Mueller. Those people denied that Trump had contemporaneous knowledge of the meeting on more than 15 occasions, according to CNN's analysis. "I don't want to use his exact words, but everything that was going on, Trump knew about".
There is no recording of the alleged meeting, CNN and NBC reported.
But perhaps the highest-stakes denial was given by Trump Jr.in his testimony a year ago to the Senate Judiciary Committee.
Trump Jr. has downplayed the meeting as a "wasted 20 minutes" and asserted Trump was never told about it.
"He was very taken aback that no one was communicating with him", Dietl said.
Trump's critics have long doubted these denials. But he also left open the possibility that Trump might have known but later forgot about it. He again said the tape does not show Trump breaking the law. The Russians then dispatched the delegation that visited Trump Jr.in NY in June with that aim. Bannon's comments, to author Michael Wolff for his book "Fire and Fury", triggered the bitter public divorce between Bannon and Trump in early 2018.
Sources tell CNN numerous tapes include discussions with reporters and conversations relating to President Trump.