(CNN) White House counsel Don McGahn's attorney did not give President Donald Trump's lawyers a full accounting of McGahn's interviews with the special counsel's team, a person familiar with the matter tells CNN.

Trump's attorneys did not ask for a complete debriefing, the person said.

During his three interviews with special counsel Robert Mueller's team, McGahn did not provide information that would be incriminating to the President, a source told CNN.

The New York Times first reported Sunday that Trump's legal team does not know the extent to which McGahn has been cooperating with special counsel Robert Mueller. The news follows a Saturday report by the Times that McGahn had been cooperating with Mueller extensively. The President's lawyers told the Times on Sunday that they believed that McGahn's nearly 30 hours of interviews with the special counsel had not included any testimony harmful to the President.

The President's advisers are now concerned McGahn's statements to the special counsel could be key to filling in some blanks of Mueller's report, two people familiar with the discussion told the Times.

According to the Times' Saturday report, McGahn's decision to cooperate was partly due to the fact that the President's initial legal team had decided to fully cooperate with Mueller's investigation into Russian interference in the 2016 presidential election, believing their client had nothing to hide and that they could bring a quick end to the probe.

McGahn and his attorney, William Burck, became concerned that the President planned to set him up to be held responsible for any potential illegal incidents of obstruction, the Times reported, so the White House counsel and his attorney came up with a strategy to cooperate as extensively as possible with the special counsel in order to prove that there was no wrongdoing by McGahn, the newspaper reported.

When contacted by CNN, Burck declined to comment on the Times' report on Sunday. Burck said in a statement Saturday, "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."

Trump's former lead personal attorney, John Dowd, who supported the idea of White House officials cooperating with the Mueller probe, told CNN on Sunday that McGahn "was a strong witness for the President's case. The strategy is working. Ask Rudy," a reference to Trump's current lead lawyer in the special counsel investigation, Rudy Giuliani.

"We protected [the President] by not asserting attorney-client privilege," Dowd added, saying that if Trump's legal team had not allowed for White House officials like McGahn to undergo voluntary interviews with Mueller's team, then they would have been called to testify before a grand jury."

Giuliani said on NBC's "Meet the Press" on Sunday morning that he had not debriefed McGahn , but nevertheless the legal team had "a good sense" of what McGahn told Mueller's team through Dowd.

"Dowd has a good sense of it, he talked to them at the time," Giuliani replied, later adding, "I think that through, through John Dowd, we have a pretty good sense of it."

"John Dowd yesterday said -- I'll use his words rather than mine -- that McGahn was a strong witness for the President, so I don't need to know much more about that," Giuliani said.

People familiar with Trump's thinking told the Times that his realization that he was unaware of what McGahn shared with Mueller's team shook the President, who is said to be obsessed with the informant role that John W. Dean, then the White House counsel to President Richard Nixon and now a CNN contributor, played during Watergate.

On Sunday morning, Trump referenced Dean in a Twitter post lashing out at the Times' report a day earlier, calling it "a Fake piece" that implied "because White House Councel Don McGahn was giving hours of testimony to the Special Councel, he must be a John Dean type 'RAT. But I allowed him and all others to testify - I didn't have to."

Dean fired back, "@realDonaldTrump (I still have trouble using the title Mr. President for someone installed by Putin), I doubt you have ANY IDEA what McGahn has told Mueller. Also, Nixon knew I was meeting with prosecutors, b/c I told him. However, he didn't think I would tell them the truth!"

"I reached out to my old friend John Dean because of what he went through with Watergate, and I saw some parallels to what Michael Cohen is experiencing," Davis told Politico. "I wanted to gain from John's wisdom."

Davis added that he had asked Dean for a retelling of his experiences during the Nixon investigation, not for legal counsel on Cohen's case.