Cohen's lawyer released a tape earlier this week of what is said to be a secretly recorded conversation between Cohen and Trump in September 2016 in which the two men discuss how to buy the rights to a former Playboy model's story of an affair with Trump.
On Friday morning, the president reiterated that claim.
"These tapes completely demonstrate the president did nothing wrong", Giuliani said on "Fox News Sunday".
It's the latest fracture between Cohen and his longtime client.
U.S. president Donald Trump has denied that he knew in advance about a Trump Tower meeting in June 2016 between a Russian lawyer, his eldest son and other aides which had been convened to uncover dirt on his Democratic rival, Hillary Clinton.
The meeting at Trump Tower in New York City involved Trump's son, his son-in-law Jared Kushner, then-campaign chairman Paul Manafort and influential Russian lawyer Natalia Veselnitskaya.
The payment to Ms McDougal was never made, according to Mr Trump's current lawyer Rudy Giuliani.
Further, Giuliani said the discussion is not on tape.
By Cohen's account, Trump approved going ahead with the meeting with the Russians, CNN reported citing sources.
Mr. Trump on Friday vehemently repeated his denial that he knew about the meeting, which is at the centre of Mr. Mueller's probe, tweeting "NO", he "did NOT know of the meeting with my son, Don jr".
She has denied acting on behalf of Russian leadership during the meeting at Trump Tower, saying she operates "independently of any governmental bodies".
If Mueller's investigators or others could establish that Trump Jr. had been speaking with his father at the time he was organizing the Trump Tower meeting with the Russian delegation, that might bolster whatever statement Cohen was willing to make to the Justice Department.
Giuliani told Fox he has hired technicians to determine if someone tampered with the tape, which cuts off abruptly. Giuliani also said that Cohen has been described as a "pathological liar".
The Mueller team has been at work for 14 months.
Cohen recently turned to prominent attorney and lobbyist Lanny Davis, a long-time associate of Bill and Hillary Clinton, to represent him.
The Thursday reports of Cohen's claims is further evidence that Cohen could flip on Trump in the Russian Federation investigation - which would be a nightmare for Trump's legal team.
"We have an outstanding offer to them", Giuliani said.
That would create another headache Trump's defense team.
Mueller, who has led the Justice Department's probe into Russian meddling, is also looking into any links between the Russians and the Trump presidential campaign.The probe is also scrutinizing whether Trump sought to obstruct justice by firing Federal Bureau of Investigation director James Comey in 2017 and otherwise undermining other Justice Department officials.
But in the run-up to the 2016 election, it seemed clear that whatever Cohen's role had been in managing Trump's political career up until that point had diminished, sources said.
And Trump Jr. talked with the Senate Judiciary Committee about the phone calls he was making at the time he was setting up the meeting, which followed an offer of help from the Russian government via a series of intermediaries.