Michael Wolff said Mr Trump repeatedly failed to recognise the accepted term for the landmark British vote, when the pair discussed it just two weeks before it was staged in June 2016.
He also predicted that his book’s revelations – including that the President’s aides all believe he is childlike and unfit for office – would eventually force him out of the White House.
The claims came as the President again hit back at the author, tweeting: “Michael Wolff is a total loser who made up stories in order to sell this really boring and untruthful book.”
Speaking to BBC Radio 4’s Today programme, Mr Wolff said: “I interviewed Trump two weeks before the Brexit vote and he didn’t know what Brexit was.”
The author explained that he had asked the then-Republican candidate how he compared himself with “the populist movements in Europe, especially with the Brexit vote coming up”.
“And he said what?”, Mr Wolff said. “I said Brexit. He repeated “what?” Then I said, the movement in the UK to leave the European Union and then he kind of said “oh yeah, yeah, yeah, I’m for that”.
“That is the entire exchange – make of it what you will.”
The publication of Fire and Fury: Inside the Trump White House has triggered a political firestorm in America, sparking renewed speculation about the President’s future.
In the interview, Mr Wolff stated his belief that Steve Bannon, the hardline nationalist who played a key role in putting Mr Trump in the White House, was now determined to see the President removed from office.
Asked if the boss of the far-right website Breitbart viewed him as “unfit and will do what he can to stop him being president”, the author replied: “Yes.”
He said the President “hardly has a staff as most people have left the White House” and that “If there is an emergency would you trust Donald Trump with that? I don’t know.”
His “almost inexplicable” election had exposed the weaknesses in the US political system but, Mr Wolff added: “It also exposes Donald Trump himself – he has no clothes.”
On Brexit, although Mr Trump may not have recognised the term, he did – in the run-up to the referendum – say the UK would be better off outside the EU, even predicting a Leave victory.
The President has claimed he never spoke with Mr Wolff for his “phony book”, tweeting on Thursday: “Full of lies, misrepresentations and sources that don't exist.”
But Mr Wolff has stood by his account and said Mr Trump's threat of legal action to stop publication was only boosting his sales.
His central and most damning allegation is that all of the President’s aides consider him to be “like a child” because of his “need for immediate gratification”.
The author said he spent about three hours with Mr Trump during the election campaign and in the White House, as well as interviewing people who spoke with him on a daily basis.
The President has gone to his Camp David retreat, hoping to steady the ship by concentrating on policy and looming elections in talks with congressional leaders.
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