Video report by ITV News Correspondent Juliet Bremner
The Times newspaper reported the book has been leaked to the media despite efforts by the Trump administration to block its publication which is planned for June 23.
A spokesperson for Mr Trump said the book should not be published as it contains "classified" information and to do so would be "unacceptable".
Mr Trump is said to have replied: "Oh, are you a nuclear power?"
Mr Bolton makes it clear the line was not intended as a joke.
The former ambassador also wrote that at a Nato summit in 2018 Mr Trump had decided to tell allies the US would pull out of the group if other countries did not increase their defence spending.
The book claims the president told Mr Bolton in a message: "We will walk out, and not defend those who have not (paid)."
Mr Trump, however, ultimately did not make such a dramatic threat, with the US staying in the alliance.
Mr Bolton said Mr Trump also asked to help his re-election chances - and had on occasion expressed a willingness to stop criminal investigations involving companies in China and Turkey in order to "give personal favours to dictators he liked".
Speaking on Wednesday evening, US Trade Representative Robert Lighthizer said the request for help with re-election "never happened".
In the memoir, Mr Bolton said the president's impeachment charge, for which he was acquitted, had indeed been true.
Mr Bolton controversially refused to testify at the impeachment trial, however, instead saving his version of events for the memoir - a book deal reportedly worth £1.5 million.