President Trump's personal attorney, Rudolph Giuliani, proclaimed Sunday that "truth isn't truth" while attempting to explain his reluctance to have Trump sit down for an interview with Special Counsel Robert Mueller's team.

"I am not going to be rushed into having him testify so that he gets trapped into perjury," Giuliani said on NBC News' "Meet The Press." "And when you tell me that, you know, he should testify because he's going to tell the truth and he shouldn't worry, well that's so silly because it's somebody's version of the truth. Not the truth."

"Truth is truth," interrupted moderator Chuck Todd, to which Giuliani responded: "No, it isn’t truth. Truth isn’t truth."

An apparently exasperated Todd joked, "This is going to become a bad meme," before telling Giuliani, "Don't do 'truth isn't truth' to me."

"[If] they have two pieces of evidence," Giuliani shot back, "[and] Trump says I didn’t tell them and the other guy says that he did say it, which is the truth? Maybe you know because you’re a genius."

"At that point, you’re right," Todd answered. "... No, you’re right. I don’t read minds on that front."

Trump repeatedly has said that he is open to sitting down with Mueller's investigators. However, attorneys Giuliani and Jay Sekulow have cautioned against it.

Both sides have exchanged proposals for interview conditions, but no agreement has been struck. The president's lawyers also have said they would fight any attempt by Mueller to issue a subpoena to Trump.

Giuliani also accused the special counsel's office of illegally leaking to The New York Times that White House Counsel Don McGahn has been cooperating extensively with the Mueller probe. The Times published the story Saturday, drawing an angry response from the president on Twitter.

"The only other one that could've done it was McGahn," Giuliani said. "I mean, I didn't leak it to The Times and Jay Sekulow didn’t leak it to The Times, the president sure as heck didn't, so who could it be? It could be McGahn, and McGahn’s not doing it. He would've done it a long time ago if he was going to do it.

"They’re down to desperation time," Giuliani said of the Mueller team. "They have to write a report and they don’t have a single bit of evidence."

Earlier Sunday, Trump tweeted that McGahn was not, as he put it, "a John Dean type 'RAT,'" -- a reference to the White House counsel for Richard Nixon during the Watergate scandal. Dean eventually cooperated with prosecutors and congressional investigators, providing key testimony that led to Nixon's resignation in the face of certain impeachment and removal from office in 1974.

Giuliani told NBC that Trump didn't raise executive privilege or attorney-client privilege during those interviews because his team believed — he says now, wrongly — that fully participating would be the fastest way to bring the investigation to a close.

"The president encouraged him to testify, is happy that he did, is quite secure that there is nothing in the testimony that will hurt the president," Giuliani said.

McGahn's attorney William Burck added in a statement: "President Trump, through counsel, declined to assert any privilege over Mr. McGahn's testimony, so Mr. McGahn answered the Special Counsel team's questions fulsomely and honestly, as any person interviewed by federal investigators must."

The Associated Press contributed to this report.