Trump attorney Rudolph W. Giuliani refuted speculation Sunday that White House counsel Donald McGahn had flipped on President Trump, accusing “desperate” investigators of leaking information about the attorney’s cooperation with the special counsel in order to lure the president into a perjury trap.

“They’re down to desperation time,” Mr. Giuliani said on NBC’s “Meet the Press.” “They have to write a report, and they don’t have a single bit of evidence.”

The New York Times, citing multiple unidentified sources, reported Saturday that Mr. McGahn had spent 30 hours over three days answering questions as part of special counsel Robert Mueller’s investigation into Russian election interference.

The report fueled comparisons to Nixon White House counsel John Dean’s decision to switch sides during the Watergate investigation, but former Trump attorney John Dowd said Mr. McGahn was actually a “terrific witness” for the president, dismissing speculation to the contrary as “pure fiction.”

“Pure fiction on claims of pushback by McGahn,” Mr. Dowd told The Washington Times. “Don McGahn was a terrific witness for the president. I debriefed him.”

Mr. Dowd also defended the Trump administration’s open-book legal strategy of providing extensive access to witnesses and documents. He said Mr. Mueller told him that no one lied and no documents were withheld.

“It’s worked out fine because now Mueller is empty,” Mr. Dowd said. “We’ve answered all his questions. … And he’s said the president has no exposure.”

Mr. Giuliani accused the special counsel’s office of leaking the report to spur Mr. Trump into testifying, then setting him up for a perjury charge by having him contradict statements by former FBI Director James B. Comey about former National Security Adviser Michael Flynn.

“I’m not going to be rushed into having him testify so that he gets trapped into perjury,” Mr. Giuliani said. “When you tell me that 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.”

The former New York City mayor wound up in a trap of another sort after he countered host Chuck Todd’s assertion that “truth is truth” with, “No, truth isn’t truth.”

“This is going to be a bad meme,” said a laughing Mr. Todd, while Mr. Giuliani responded, “No, don’t do this to me.”

Mr. Giuliani described a situation as a case of he-said, he-said: “Donald Trump says, ‘I didn’t talk about Flynn with Comey.’ Comey says, ‘You did talk about it.’ So tell me what the truth is.”

The Times report said Mr. McGahn decided to cooperate fully because he was worried that the president was trying to set him up to take the fall for any obstruction charges by refusing to shield their conversations with executive or attorney-client privilege.

Mr. Trump tweeted Sunday that the story was a “fake piece,” that he had “nothing to hide” and that his administration was the “most transparent in history.”

“The Failing New York Times wrote a story that made it sound like the White House Councel [sic] had TURNED on the President, when in fact it is just the opposite,” Mr. Trump said in a Sunday tweet, adding, “This is why the Fake News Media has become the Enemy of the People. So bad for America!”

In a statement, the New York Times communications department said the newspaper “stands behind the reporting of our Pulitzer Prize-winning reporters.”

McGahn attorney William A. Burck released a statement confirming that his client had cooperated fully.

“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,” said the statement.

Mr. Giuliani said the president has “encouraged all the people who testified to tell the truth, to take as long as they needed to do that, and as long as they did, they’ll have no problem with the president or us.”

“And we have no reason to believe that Don McGahn didn’t do that,” he added.

He said the leak indicated that the Mueller team is panicking because “they know they don’t have a case, there was no collusion, there was no obstruction, they can’t prove it, and they are trying to get the president to testify,” Mr. Giuliani told Fox’s “Justice with Judge Jeanine.”

“And they’re hoping that if they put out a story like this in which they suggest that McGahn is cooperating against him — but don’t say it, they don’t say that — that he’ll want to come in and explain himself,” said Mr. Giuliani.

He said the president has turned over 1.4 million documents, encouraged more than 30 witnesses including Mr. McGahn to testify, and declined to assert executive privilege as part of his commitment to transparency.

Mr. Trump’s decision to decline to assert executive and attorney-client privilege has come into question, but Mr. Dowd said “it was smarter to be transparent and get the case over with.”

“We made available 37 witnesses and a million documents,” said Mr. Dowd. “In terms of McGahn, he’s right at the heart of the case and they had a lot to cover with him and a lot of notes. It was all routine and very normal. The anonymous sources that criticize us for our strategy don’t realize that the alternative was a nightmare, just a war.”

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