The book portrays Trump as an ill-informed and dishonest president and the West Wing staff as trying to minimize the damage.
"You can, in an nearly microscopic way, establish what occurred - and that's what I've done in this book".
The Watergate legend said the president called him last month saying he wished he had been interviewed for the book.
But Woodward insisted his book was accurate and said he carefully vetted each description included in the book.
While neither former staffer directly denied details in the book, Porter said in a statement that the book offers a "selective and often misleading portrait".
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".
The president blasted Brzezinski as "low I.Q. Crazy Mika" and said she had been "bleeding badly from a face-lift" during a New Year's party at his Florida resort.
Trump also defended former Republican Senate candidate Roy Moore, who was running for an Alabama vacancy and was accused of engaging in sexual misconduct with teenagers when he was in his 30s.
Woodward stands by his reporting, telling NPR he attributes the denials to "political necessity".
After the chemical attack on Syrian civilians in April 2017, which prompted outrage across the world, Mr Trump reportedly said: "Let's f****** kill him!" Trump asked, according to Woodward.
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.
"There's a war on truth", Woodward's answered, echoing what we've heard before. Hall said he believes the current landscape is not amendable to impeaching Donald Trump, but that would change if Democrats regain control of the House and Senate. "Frankly, the White House and the staff here are focused on doing our jobs", she told CNN's Jeff Zeleny.
Alas, that was before the president came to the conclusion that Woodward was working for the Dems and that he made quotes up out of whole cloth, which was about the same time that excerpts of Woodward's book began to leak.
The White House says Camp David, the USA president's country retreat, offers the commander-in-chief "solitude, tranquility, and a place to host foreign leaders".
"As anyone who has worked for Trump knows and I think that establishes this also", Woodward told Perino, "that it's very hard to control Trump".