As White House counsel since the beginning of the Trump administration, McGahn could have rare insight into the president's thinking.
"It is not clear that Mr. Trump appreciates the extent to which Mr. McGahn has cooperated with the special counsel", the Times reported.
Trump confirmed that he gave McGahn and other White House staffers permission to fully cooperate with the investigation, which he has regularly dismissed as unsubstantiated and unfair.
Giuliani said he did not believe McGahn was cooperating against the president, noting Trump's lawyers and McGahn's have a joint defense agreement that would have otherwise ended.
Giuliani said that discussions over a presidential interview continue with Mueller's office and that he would not be rushed into having Trump testify "so that he gets trapped in perjury".
Brennan is also scheduled to appear on NBC's "Meet the Press" on Sunday.
US President Donald Trump has taken to Twitter to launch another scathing attack on the New York Times and Special Counsel Robert Mueller's Russian Federation probe, calling it "McCarthyism at its WORST".
William Burck, McGahn's attorney, provided Trump's lawyers with a brief overview of McGahn's initial interview with the special counsel in November but hasn't told them what McGahn has said in further conversations with the Mueller's team, according to the newspaper.
According to the New York Times, McGahn in at least three voluntary interviews with investigators that totaled 30 hours over the past nine months, described Trump's furor toward the Russian Federation investigation and the ways in which the president urged McGahn to respond to it.
Trump says he authorized White House staff to cooperate with Mueller
He added in response to Mr Trump's tweets that he doubts the president has "ANY IDEA what McGahn has told Mueller".
The president tweeted Saturday evening that he "allowed White House Don McGahn, and all other requested members of the White House Staff, to fully cooperate with the Special Counsel". The former FBI director James Comey (fired by Trump in an event looked at by Mueller regarding potential obstruction of justice), fired deputy FBI director Andrew McCabe and fired FBI agent Peter Strzok are among those reported under threat of similar treatment. "So, I think this is really a question of impression, not of anything that we have to worry about".
Mr. Trump repeated that there's been "no collusion and no obstruction" on his part, and instead blamed "crooked" Democrats including Hillary Clinton as well as former members of the intelligence community.
McGahn cautioned to investigators he never saw Trump go beyond his legal authorities.
On Sunday, he compared Mueller with 1950s-era USA senator Joseph McCarthy, whose anti-Communist crusade eventually led to his censure by the Senate.
While the claims seemed quite stunning at the time, many people still believed McCarthy.