The writer, claiming to be part of the "resistance" to Trump but not from the left, said, "Many Trump appointees have vowed to do what we can to preserve our democratic institutions while thwarting Mr. Trump's more misguided impulses until he is out of office".
Mr Woodward's much-awaited book, Fear: Trump In The White House, shows aides routinely ignoring or acting to block orders by the US President.
In the forthcoming chronicle of Trump's White House, "Fear", Woodward writes that the president privately called Sessions a "traitor", saying: "This guy is mentally retarded".
"The contemptuous words about the President attributed to me in Woodward's book were never uttered by me or in my presence", Mattis said in a statement, charging that the journalist's "anonymous sources do not lend credibility".
In a transcript and audio of the call released by the Post, Woodward tells Trump he made repeated efforts to get an interview, while a clearly irritated Trump says he would have participated if he had known. For this book, Woodward spoke to top aides and other insiders with the understanding that he would not reveal how he got his information, the Post reported.
The article reportedly caused the president to erupt with "volcanic" anger, two people familiar with his reactions told the Post, and that fury was barely contained in public as Trump raged against what he called a "gutless editorial". "We want the administration to succeed and think that many of its policies have already made America safer and more prosperous".
The opinion piece pointed out that the Trump administration has pushed ahead with sanctions against North Korea and Russian, even while the president himself had expressed great admiration for Kim Jong-un and Vladimir Putin.More news: Colin Kaepernick's Nike ad is naturally a meme now
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Mueller has asked the White House for access to Trump, and administration officials have negotiated for months over how that might happen.
Trump also tweeted out written statements by aides on the Woodward book, including one from his chief of staff.
Late Tuesday night, Trump tweeted links to stories by The Daily Caller and Breitbart News that highlighted Mattis' denial of the claims in Woodward's book. "What you're seeing and what you're reading is not what's happening", he told supporters earlier this year. It also says he called Trump an "idiot", an account Kelly denied Tuesday.
Woodward's book, set to be released next Tuesday, comes from hundreds of interviews with participants and witnesses, along with meeting notes, personal diaries, and government documents. "Look, all of this stuff is obviously insane, but at some point I think we've got to stop saying that it's a 'bombshell, '" he said.
Regarding the fraught relationship between the president and his chief of staff, the Post reported Woodward found that Kelly "frequently lost his temper" over Trump, and that he thought the president was "unhinged". I think Sessions is a very smart man and a man of integrity.
"Mr. Woodward never discussed or verified the alleged quotes included in his book with Secretary Mattis" or anyone within the Defense Department, Manning said.
Press secretary Sarah Sanders says in a statement: "This book is nothing more than fabricated stories, many by former disgruntled employees, told to make the President look bad".
A central theme of the book is the stealthy machinations used by those in Trump's inner sanctum to try to control his impulses and prevent disasters, both for the president personally and for the nation he was elected to lead.