Tillerson's comments came in an interview to CNN in the aftermath of a bombshell book that cited White House aides questioning the president's ability to serve.
"I've never questioned his mental fitness, I've had no reason to question his mental fitness," Tillerson said.
Anecdotes in the controversial book about Trump's alleged inability to follow briefings or read notes recalled reports last year that Tillerson once called the president a "moron."
The State Department denied that claim, but ever since Tillerson has been dogged by reports that White House officials close to Trump are scheming to oust him.
But Tillerson told CNN that Trump has never sought to push him out and that he himself plans to remain.
"I intend to be here for the whole year," he said. Asked whether the president had given him any indication that he wants him to step down, Tillerson said: "None whatsoever."
Trump says book is 'phoney' and 'full of lies'
The release of a bombshell-filled book about Donald Trump's first year in the White House on Friday sparked fresh debate about the president's fitness for office, with the author claiming his closest aides "say he is like a child."
"Fire and Fury: Inside the Trump White House" was rushed into bookstores and onto e-book platforms four days ahead of schedule due to what its publisher called "unprecedented demand" -- and after Trump's bid to block it failed.
The book -- which has sent shockwaves across Washington -- quickly sold out in shops in the US capital, with some even lining up at midnight to get their hands on it. Trump has decried the instant best-seller as "phony" and "full of lies."
US President Donald Trump is in a public relations crisis after the release of new book 'Fire and Fury: Inside the Trump White House'. Source: Getty Images
Journalist Michael Wolff, no stranger to controversy, quotes several key Trump aides expressing serious doubt about his ability to lead the world's largest economy -- and despite fiery criticism from the Republican, he stood his ground.
"Let me put a marker in the sand here. One hundred percent of the people around him" question Trump's fitness for office, Wolff said in an interview with NBC's "Today" show.
"They all say he is like a child. And what they mean by that is he has a need for immediate gratification. It's all about him."
The 71-year-old Republican president, who is approaching the first anniversary of his inauguration, has responded with fury to the claims in Wolff's book.
"I authorized Zero access to White House (actually turned him down many times) for author of phony book! I never spoke to him for book. Full of lies, misrepresentations and sources that don't exist," Trump tweeted Thursday.
But Wolff countered in Friday's interview: "I absolutely spoke to the president. Whether he realized it was an interview or not. I don't know, but it certainly was not off the record."
The book -- which paints Trump as far out of his depth -- includes extensive quotes from Steve Bannon, his former chief strategist, and his publication sparked a very public break between the former allies.
Bannon is quoted accusing Trump's eldest son Don Jr of "treasonous" contacts with a Kremlin-connected lawyer, and saying the president's daughter Ivanka, who imagines running for president one day, is "dumb as a brick."
Criticism from aides
But it is Trump himself who is cast in the most unfavorable light by a series of his top aides.
The book claims that for "Steve Mnuchin and Reince Priebus, the president was an 'idiot.' For Gary Cohn, he was 'dumb as shit.' For H.R. McMaster, he was a 'dope.' The list went on."
At least a dozen members of the US Congress, most of them Democrats, were briefed by a Yale University professor of psychiatry on President Donald Trump's mental health, US media reported Thursday.
The briefing by Dr. Bandy Lee, the editor of "The Dangerous Case of Donald Trump: 27 Psychiatrists and Mental Health Experts Assess a President," took place in early December.
"Lawmakers were saying they have been very concerned about this, the president's dangerousness, the dangers that his mental instability poses on the nation," Lee told CNN.
The White House issued a scorched-earth dismissal of "Fire and Fury", its author and his sources, with press secretary Sarah Sanders, calling the book "complete fantasy."
Behind the scenes, though, Trump has been enraged by the betrayal by Bannon -- a man who engineered the New York real estate mogul's link to the nationalist far right and helped create a pro-Trump media ecosystem.
Sanders suggested that Bannon's employer, Breitbart News, should consider firing him.
He wasn't fired, but Bannon's main financial backer is formally cutting ties with him, The Washington Post reported.
US President Donald Trump congratulates Senior Counselor to the President Stephen Bannon during the swearing-in of senior staff in January 2017. Source: AAP
"I support President Trump and the platform upon which he was elected," the newspaper quoted billionaire conservative donor Rebekah Mercer as saying.
Bannon, who left the White House in August, is also quoted in the book as saying that the investigation by special counsel Robert Mueller into Russian interference in the 2016 election will focus on money laundering.
The investigation by Mueller, a former FBI director, is looking into whether the Trump campaign colluded with Russia to help get him elected -- a charge the president has repeatedly and vehemently denied.
Disbelief over victory
Wolff said his account was drawn from interviews with those in close contact with Trump and all described him in the same terms.
"They say he's a moron, idiot," Wolff told NBC.
"Actually, there's a competition to sort of get to the bottom line here of who this man is. Let's remember, this man does not read. Does not listen. So he's like a pinball. Just shooting off the sides."
And the author confidently defended himself against attacks on his credibility, which have included threats from Trump's lawyers of a libel suit.
"My credibility is being questioned by a man who has less credibility than, perhaps, anyone who has ever walked on earth at this point," Wolff said.
"I spoke to people who spoke to the president on a daily, sometimes minute-by-minute basis," he added, saying he had notes and recordings of the interviews.
"I am certainly absolutely in every way comfortable with everything I've reported in this book."
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