Woodward, whose new book "Fear: Trump in the White House" hits book stores on Tuesday, described the incident in the interview with CBS as the most unsafe moment of Trump's nuclear standoff with North Korea.
"The people are not anonymous to me", he says.
In particular, he highlighted an instance in which Trump complained to Defense Secretary James Mattis and other staff at an NSC meeting about the expense of US forces overseas.
According to the network, Woodward writes that in one such instance then-National Economic Council Director Gary Cohn stole a letter from Trump's desk that would have led to the termination of the free trade agreement between the US and South Korea.
"Just another assault against me, in a barrage of assaults, using now disproven unnamed and anonymous sources", Trump said.
At a National Security Council meeting on January 19, the president "disregarded the significance of the massive US military presence on the Korean Peninsula, including a special intelligence operation that allowed the United States to detect a North Korean missile launch in seven seconds vs. 15 minutes from Alaska", The Post reported.
Woodward said he used multiple interviews with key witnesses to write the book, including one subject who was interviewed nine times, resulting in hundreds of pages of their transcribed statements.
Mattis, Kelly and other senior aides have denied making the remarks.
As part Trump's efforts to discredit Woodward on Monday, he wrote another tweet in which he quoted Washington Examiner reporter Katelyn Caralle as saying, "You can not take this book too seriously".
That's a big deal - especially because it confirms much of the daily reporting that has come out of this White House since Trump became President 19 months ago. Dems can't stand losing.
"These are political statements to protect their jobs", Woodward said. The anonymous writer goes on to denounce Trump's "mass-marketing of the notion that the press is the 'enemy of the people, ' and says his "impulses are generally anti-trade and anti-democratic".
Insider accounts of the Trump White House make a ton of money.
Trump responded to the book Wednesday in the Oval Office and said, "The book is fiction".
According to Woodward, Porter asked the president if the tweet might provoke Kim, to which he replied "it's leader versus leader".
"There's clearly a rising Republican tide of opposition to Trump, and he will not handle that well", said Mike Murphy, a Republican consultant who opposed Trump's candidacy in 2016. "I don't know if that's in the same class of undercutting the president's agenda but it's certainly trying to reign in" the president, Flake said. "Family members of the 28,000 people there, '" he said.