In the book, Mr Bolton claims that Mr Trump does not have the competence to be the US President.
"You were given the chance to testify at his impeachment, you were begged to do your duty as a citizen, you refused to tell the American people what you saw and what you knew," Ms Maitlis said.
Mr Bolton said the Newsnight presenter was "absolutely wrong about that", leading to a tense exchange.
"Well you're absolutely wrong about that, there's a history here that I'd be delighted to lay out, although I'm sure you won't broadcast any of it because it's too complicated," he said.
Ms Maitlis interrupted: "You were asked to testify".
Mr Bolton then told her to "let him finish" several times.
He said: "Will you let me finish? Will you let me finish my answer?"
He then explained: "The House of Representatives never issued a subpoena for me. And when they issued a subpoena for my former deputy, the White House ordered him not to testify."
Ms Maitlis also asked Mr Bolton why he would work for Mr Trump if he saw him as corrupt and a threat to US security.
Mr Bolton joined the White House in April 2018 and left in September the following year, saying he had decided to quit as national security adviser. because he disagreed "strongly" with him.
He replied: "I was worried about him endangering American security that's why I tried to move his policies in a better directions and why ultimately when I concluded that I couldn't do that, I resigned."
Democrats charged Mr Trump in December with pressuring Ukraine to smear a potential White House rival.
But he was found not guilty in the impeachment trial, ending a bid to remove him from office that bitterly divided the US.
The Senate, run by the president's fellow Republicans, voted to acquit him 52-48 on charges of abuse of power and 53-47 on obstruction of Congress.