The book portrays President Donald Trump as erratic and impetuous, and surrounded by a staff trying to restrain him from his worst instincts.
Priebus wouldn't have any of it and said there will be a dinner in the White House.
Trump has attacked the book and author, calling it "fiction" and Woodward an "idiot".
"You've got to deny, deny, deny and push back on these women", he told the person, who was not named, Woodward reported. They've denied making the statements.
In recent days, Trump has accused Woodward of making up quotes and mused about changing US libel laws in an effort to better position himself to seek "retribution".
Mr Woodward told the BBC: "People in the White House they told me, to a certain extent, they spend a third of their time stopping things that are unsafe or not really consistent with a policy to protect the United States and the national security".
"Fear: Trump in the White House" hits U.S. bookstores on Tuesday after a weeklong buildup, with published excerpts, leaks and interviews that portray Trump as dangerously erratic and uninformed. Woodward attributed to him were "incorrect" and that he hadn't been called by Mr.
Amid the drama, former staffer Omarosa Manigault Newman sought to seize some of the spotlight by releasing a recording she said was of Trump talking to junior aides during a communications meeting.
The op-ed was not the only thing on Trump's mind as he called around to advisers over the weekend, according to two people familiar with the conversations.
How to Watch Apple's iPhone Event
Apple may or may not reveal the new laptop Wednesday , when it is holding a press event on its campus in Cupertino, California. If so, that would mean the home button, the circular button that made the iPhone instantly recognisable, will be no more.
Among the revelations in the book: Cohn once removed a document from the president's desk that would have ended a trade deal with South Korea.
"We're doing this in order to prevent World War III", Mattis told the president, according to Woodward's reporting.
Woodward explains: McMaster, he says, "understood the strategic importance of India, a sworn enemy of Pakistan".
"She's obsessed with Barron", one person told Woodward.
How times have changed since the says when Woodward was instead at odds with President Barack Obama's White House.
Bob Woodward, associate editor of the Washington Post, has recently published an explosive book on the current USA presidency. "People closest to him do not trust him", he added, "And the impulse-driven presidency is something that the more people know, the more they realize we're at risk".
At a White House press briefing, Sanders slammed the book "Fear", from veteran journalist Bob Woodward, as "careless and reckless".
Trump has already tweeted his low opinion of the book.
- Kerala nun writes to Vatican envoy against Jalandhar diocese bishop
- Bezos calls Trumps attacks on media 'dangerous'
- Aaron Rodgers, Packers taking knee injury 'day by day'
- Apple Watch 4 wants to keep you fitter and more alive
- Arvind Kejriwal calls Mallya's claims on meeting Jaitley 'absolutely shocking'
- California aims to get all electricity from clean sources
- Senate Judiciary Committee delays Kavanaugh vote until next week
- England players 'harsh on themselves' at half-time vs. Switzerland - Gareth Southgate
- Incredible video shows plane flying into eye of Hurricane Florence
- White House adviser admits Trump got his facts wrong in economy tweet