When interviewer Chuck Todd shot back "Truth is truth", Giuliani replied, "No, it isn't truth".
Todd said this was going to become a "bad meme".
If Mueller tries to issue a subpoena to Trump before the midterm elections in November, the president's team will accuse him of trying to interfere because he could have acted sooner, Giuliani said on NBC.
Mr Giuliani said: "I'm not going to be rushed into having him testify so can he can be trapped into perjury".
Todd interrupted. "Truth is truth. So tell me what the truth is", Giuliani said.
The rationale used by Trump and now by Giuliani contradicts a statement the president helped craft in 2017, which said the meeting with lawyer Natalia Veselnitskaya had been arranged to discuss USA policies for adopting Russian children. Giuliani insisted, as he has on previous occasions per CNN, that Mueller would attempt to set a "perjury trap" for Trump if Trump were to testify in the Special Counsel's investigation into collusion between the 2016 Trump campaign and Russian Federation.
"I can figure it out", Giuliani said of the substance McGahn's interviews.
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This is not the first time that Mr Giuliani, a former New York City mayor, has spoken of irreconcilable accounts of the same event.
"It's somebody's version of the truth, not the truth", he said, telling NBC's Chuck Todd: "Truth isn't truth".
He pleaded guilty to lying to the Federal Bureau of Investigation and is cooperating with Mueller's probe.
Elsewhere in the interview, Giuliani executed the same 180° turn the President did weeks ago, acknowledging that the campaign-era meeting in Trump Tower between senior campaign officials and Russians "was originally for the goal of getting information about [Hillary] Clinton". The Times published the story Saturday, drawing an angry response from the president on Twitter.
This interview comes amid the everlasting dance between the Trump administration and Mueller's team.
Rudy Giuliani, the lawyer and public spokesperson for Donald Trump, tried out a unique new defense of Trump on "Meet the Press" Sunday. "If someone said, I have information about your opponent, you would take that meeting". They have taken two to three weeks to get back to us.
Critics of the president have recently compared his decision to target the security clearances of critics and those involved in the Russian Federation investigation to the "enemies list" created by the Nixon White House to keep track of political opponents they meant to target with punitive measures.