He alluded that Cohen was "bitter" about not landing a job in the Trump administration and was "very, very jealous" of the president's children.
Authorities in the government's investigation of Cohen are reportedly analyzing multiple recordings.
"We have complained" to Cohen's lawyers that "he's violated the attorney-client privilege, publicly and privately", Giuliani told the network. That turned out to be misleading at best after revelations that the meeting's organizer had pledged damaging information from the Russians on Clinton but the participants ended up discussing relief from USA sanctions on Russia.
The recording is unclear about what then-candidate Trump is saying, but his lawyer, Rudy Giuliani, and a transcript of the recording compiled by Trump's legal team, maintains that he is saying, "Don't pay with cash ... check".
It may be this information that the legal team warned Cohen about, though Giuliani wasn't clear in his ABC interview. Cohen does not have any evidence such as audiotapes verifying his claims, CNN's sources said.
On that recording, released by Cohen's attorney Lanny Davis, Trump can be heard discussing with Cohen arrangements for payment of hush money to Playboy model Karen McDougal, who says she had an affair with Trump in 2006.
Trump denied knowledge of the meeting once again in a blistering tweet directed at Cohen.
"Sounds to me like someone is trying to make up stories in order to get himself out of an unrelated jam", Trump tweeted, adding: "(Taxi cabs maybe?).
Cohen reached out to ex-Trump aide Steven Bannon, according to the report, who has, like Cohen, been exiled from the president's inner circle.
(FILES) In this file photo taken on September 26, 2016, Republican presidential nominee Donald Trump (R) stands with his son Donald Trump Jr. after the first presidential debate at Hofstra University in Hempstead, New York.
The US leader's latest Twitter outburst came after Russian President Vladimir Putin revealed he has issued an invitation to Mr Trump for a face-to-face meeting in Moscow.
Alan Futerfas, a lawyer for the Trump Organization and Donald Trump Jr., told Reuters, "Donald Trump Jr. has been professional and responsible throughout the Mueller and congressional investigations".
Trump Jr. told Senate investigators his father "wasn't aware of it".
I suspect that a long line of presidents would dispute Trump's "tougher on Russia" claim, particularly Ronald Reagan. Second, Trump could have known about the meeting and Trump Jr. may not have been the source of that information. Trump Jr. says he didn't get any dirt at the meeting - and the promised speech never happened.
"The only Collusion with Russian Federation was with the Democrats, so now they are looking at my Tweets (along with 53 million other people) - the rigged Witch Hunt continues!"