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".
Others agreed that the book is suspect. He attempted to sue Bannon and Michael Wolff and the book publisher to get the book not published.
The publication came as it emerged that at least a dozen members of the US Congress were briefed last month by a Yale University professor of psychiatry on Trump's mental health.
"I authorized Zero access to White House (actually turned him down many times) for author of phony book!"
"I don't describe him as childlike, every person in the White House".
Bannon's reaction to the controversy has been muted.
In a tweet on Friday, Mr Trump called the book another attempt to smear him since any collusion between his election campaign and Russia "is proving to be a total hoax".
The White House has dismissed the book as "tabloid gossip" full of "lies" and some commentators have cast doubt on Wolff's credibility.
Trump said earlier this week that Bannon had "lost his mind". Some lines in some areas of the country went around the block.
It already has cost him a key backer.
"Once the spin starts you get this cloud surrounding everything, and it becomes a media frenzy", Mahaffee said in a VOA interview.
Wolff told NBC's Today show the president is wrong.
During the meeting in December, lawmakers said they were worried about his mental state. He just has to be satisfied in the moment. On Friday, a longtime friend and confidant of Trump rejected Wolff's suggestion. Whether he realised it was an interview or not.
"I wasn't gonna buy the book. He was remembering things, he was on point, he was following up on discussions". "Nothing found. This is McCarthyism!" That only added to the frenzy at bookstores where it was available early Friday, several days ahead of schedule. His ultimate goal, after all, had never been to win.
He is not a typical President of the past, I think that's well-recognised.
Amid such backlash, Steve Bannon, Trump's campaign chief executive and former White House strategist who is a major source for Wolff's book, appears increasingly isolated. "His spot-on portrait of Trump as an emotionally erratic president, and the low opinion of him among some of those serving him".
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