He praised Trump's legislative accomplishments, which he said Woodward's book ignores, and said the president "invites robust discussion" from aides.
"You can, in an nearly microscopic way, establish what occurred - and that's what I've done in this book".
The book, which has been described by the White House as a work of fiction, has a small segment on India and June 26, 2017 visit of Modi to the White House.
Still, Woodward's book has fed talk that Trump could seek to replace Kelly and Mattis.
"That's fine", Cohn said, Woodward reports.
Sanders said it's up to the Justice Department to make that determination.
Once he won, Trump considered an unusual approach that was quickly slapped down by his chief economic advisor, according to veteran journalist Bob Woodward's new book, "Fear: Trump in the White House", which went on sale Tuesday.
The Washington Post editor and author of Fear: Trump in the White House recounted an incident in which the president complained about USA troops in the Korean peninsula.
Arun Jaitley slams at Mallya for false claim
In an 8-minute long video accessed exclusively by News18, pictures of barrack number 12 of Arthur Road jail in Mumbai were leaked. He went on to claim that the BJP had proof that the Gandhi family was helping Vijay Mallya with "sweet" deals.
Woodward stands by his reporting, telling NPR he attributes the denials to "political necessity".
And the USA ambassador to the United Nations, Nikki Haley, said that she had never heard Mr Trump talking about assassinating the Syrian president.
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".
Woodward wrote that those who traveled with Trump regularly noticed that the couple shares a honest affection, though Melania operated independently from Trump.
Then-economic adviser Gary Cohn reportedly told Trump that he couldn't raise the top tax rate for the simple reason that he was Republican and would "get absolutely destroyed" if he did so. "Never has a country given so much away for so little in return" as the U.S. had in Afghanistan, Modi had said to Trump, according to The Washington Post.
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".
"(Defense Secretary) James Mattis says we're doing this to prevent World War III", Woodward added.
In his interview with NPR, Woodward also addresses the anonymous op-ed published in The New York Times shortly after the first excerpts of his book were circulating. "I don´t even know why any of us are here".
"If that person had come to me while I was working on this book or if it were someone I interviewed I would say, 'I need specifics.' The building blocks of journalism, of truth, are specific incidents".