"I wouldn't have used it", Woodward added of the Times opinion column during his first televised interview about his 19th book, "Fear: Trump in the White House", which is scheduled for release on September 11.
During the same meeting Trump reportedly asked C Boyden Gray, former United States ambassador to the EU: "Which is going to be the next country to drop out [of the EU after Brexit]?"
"I'd do it under certain circumstances", Trump told reporters on board Air Force One.
According to the book Rear Admiral Ronny Jackson, Trump's doctor until earlier this year, repeatedly advised John Kelly, his chief of staff, to "dial back" the president's schedule because he was "under stress". It is the latest book to detail tensions within the White House during Trump's 20-month-old presidency.
Those sentiments landed on a hot public griddle, courtesy of famous presidential chronicler Bob Woodward.
"That people better wake up to what's going on", Woodward said.
The first lady had a specific message for the author: "To the writer of the op-ed - you are not protecting this country, you are sabotaging it with your cowardly actions".
In an interview with CBS News, the veteran journalist teased a scene from his upcoming book 'Fear: Trump in the White House' that illustrates just one example of the President doing something that could "threaten the national security or the financial security of the country, or of the world".More news: Including children - in Perth
Woodward has said he stands by his reporting.
Pence said on CBS's "Face the Nation" that he believes the denials from multiple Trump officials who've said they didn't write the New York Times piece, adding that the author "doesn't really know what happens in the White House".
Experts said it was unlikely the Justice Department would have sound legal grounds to get involved over a hunt for the op-ed author, unless the person was a member of the military, who are forbidden to undermine or defame the commander in chief.
Although the writer says they support the administration's objectives, its successes have come in spite of the president, who is described as impulsive, erratic and amoral, someone whose "misguided impulses" need to be controlled for the good of the US.
"You look at the operation of this White House and you have to say, 'Let's hope to God we don't have a crisis, '" Woodward told CBS.
"There's virtually no context in which this kind of op-ed comes within a mile of federal criminal law", said Stephen Vladeck, a law professor at the University of Texas.
Woodward on Sunday brushed off Trump's attacks. In another supposed episode, Woodward wrote that former top economic adviser Gary Cohn planned to remove a memo from the president's desk that would have withdrawn the United States from the North American Free Trade Agreement with Mexico and Canada.