Wolff's book, "Fire and Fury - Inside the Trump White House", portrays Trump as unfocused, unprepared and petty while presiding over a chaotic White House.
Trump has repeatedly invoked Ronald Reagan, tweeting Sunday that the former president "had the same problem and handled it well".
He went on: "I don't know this man".
He said his journey from "Very successful businessman", to reality TV star to president on his first try "would qualify as not smart, but genius. and a very stable genius at that!"
Trump spoke during an informal press conference at Camp David on Saturday. "If I weren't involved they wouldn't be talking at all right now", Mr Trump said, but made clear his stance. "I had a situation where I was a very excellent student, came out and made billions and billions of dollars, became one of the top business people, went to television and for 10 years was a tremendous success, as you probably have heard, ran for President one time and won. Didn't exist, it's in his imagination".
But in a Friday interview with NBC's "Today" show, Wolff insisted that he absolutely spoke to the president. Rep. Al Green, D-Texas, said Trump had associated his presidency with causes rooted in bigotry and racism.
Behind the scenes, though, Trump has been enraged by the betrayal by Bannon - a man who engineered the NY real estate mogul's link to the nationalist far right and helped create a pro-Trump media ecosystem.
Questions about the president's mental fitness also appear in Fire and Fury: Inside the Trump White House,the salacious new book on the early months of the Trump White House by longtime media writer Michael Wolff.
It wasn't long before the book created a public rift between Trump and Bannon, who's always been one of the president's most ardent supporters.
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Instead it attempted to tie the growth to things like the App Store's big makeover and the new crop of AR apps, like Pokémon Go. That would suggest that Apple's cut of App Store sales was about $11.35 billion, based on the historical 70-30 split.
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CNN has not independently confirmed all of Wolff's assertions. He gives Trump his regular intelligence briefings. Bandy X. Lee on Capitol Hill in early December about Trump's fitness to be president.
"I consider it a work of fiction and I think it's a disgrace that somebody is able to have something, do something like that", Trump said. He should have known they are duplicitous, cunning and not our friends.
Nikki Haley, the US ambassador to the United Nations, said that she is at the White House once a week, and "no one questions the stability of the president".
The book also quotes Bannon and other prominent advisers as questioning the president's competence.
The book includes extensive quotes from Steve Bannon, Trump's former chief strategist, and its publication sparked a very public break between the former allies. Now Sloppy Steve has been dumped like a dog by nearly everyone.
"My support is also unwavering for the president and his agenda - as I have shown daily in my national radio broadcasts, on the pages of Breitbart News and in speeches and appearances from Tokyo and Hong Kong to Arizona and Alabama". "Suddenly everywhere people are going, 'Oh my God, it's true, he has no clothes.' That's the background to the perception and the understanding that will finally end. this presidency".
White House Chief of Staff John Kelly told reporters at Camp David that Mr Trump did not seem angry about the book and on Friday night had watched a new film, The Greatest Showman about legendary circus promoter P T Barnum, with Republicans and his cabinet.
He was reacting on Saturday to an author's accusations that he is mentally unfit for office.
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