Bob Woodward dismissed claims separate audio recordings would show Jared Kushner misquoted as saying former Trump administration members were "overconfident idiots".
"I report accurately," Mr Woodward said in a Washington Post interview on Tuesday.
In an appearance on the Today Show earlier on Tuesday, Mr Kushner made a veiled threat to release the audiotapes made as part of the research for Rage, Mr Woodward's inside look at the Trump White House.
Mr Kushner, Trump's son-in-law and senior adviser, is quoted extensively in the book, saying that the most dangerous people around the president are "overconfident idiots".
Watch: Jared Kushner asked to clarify comment made in Bob Woodward’s new book ‘Rage’
Mr Woodward wrote that Mr Kushner was referring to former secretary of defence General Jim Mattis, former secretary of state Rex Tillerson, and former National Economic Council director Gary Cohn.
But on Tuesday, Mr Kushner said he was "mischaracterised" by the veteran journalist and he had the recordings to prove it.
"That wasn't clear and again, he's got tapes of everything, I have tapes of everything so that was never implied in that regard," Mr Kushner said.
He added: "He actually mischaracterises who I was referring to, obviously the people that we had, some people from the campaign who were in there who were obviously trying to tell the president with confidence without the real facts."
Watch: Bob Woodward explains his reporting on Jared Kushner in new book ‘Rage’
While Mr Woodward said he recorded all his conversations with permission, the White House did not extend him the same courtesy of advance notice that the interviews were captured on tape.
He said there were far more important quotes from Mr Kushner in the final edition of the book, including the question of whether Trump executed a "hostile takeover" of the Republican party.
“I’m quite interested in, when Jared says he has tapes, I have tapes. I taped him with his permission. I suspect that he was taping me. He did not extend the courtesy to me that he was taping the conversation,” Mr Woodward said.
“That’s fine. I report accurately what he said in the book.”