Spokeswoman Sarah Huckabee Sanders said no lie detectors were being used to smoke out the writer of the op-ed, though she insisted the West Wing would like to see an investigation.
Trump then told Cohn that he wouldn't pick him to be Fed chair, according to the book.
Trump and several members of his administration were quoted have called the book "fiction and fabrication". Some former White House officials who spoke with Woodward have tried to distance themselves from the book.
"Time and time again people will deny things", Woodward says.
Earlier, Trump said he would pen his own book with the real story of his accomplishments.
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But ahead of the Raiders' first game of the season, head coach Jon Gruden had some choice words about Mack's holdout in Oakland. He had heavy pressure on Rodgers early in the second quarter which resulted in a sack by Roy Robertson-Harris.
Withdrawing from that trade agreement, known as KORUS, could jeopardize a top-secret intelligence operation that enables the U.S.to quickly detect any intercontinental ballistic missiles launched from North Korea, according to Woodward.
While the op-ed prompted furious backlash from the president and his allies, 42 percent of respondents in the poll said they feel more comfortable when administration officials openly criticize the president.
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".
Trump also derided his predecessor's return to politics. On Monday, it was ranked as the top-selling book on Amazon.
"The Woodward book is a Joke - just another assault against me, in a barrage of assaults, using now disproven unnamed and anonymous sources", Mr. Trump tweeted.
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Ms Quinn's mother was named in reports as Beverley Quinn, 74. "I want everyone just to hug their kids, hug their family today". He will appear in Perth Magistrates Court via videolink from Karratha Magistrates Court later on Monday.
"Dems can't stand losing".
The president's frustration grew when, soon after portions of Woodward's book were leaked, the New York Times published an anonymous op-ed by a senior Trump official who wrote of a quiet "resistance" inside the Trump administration, working against some of the president's wishes.
The anonymous official, identified only as "a senior official in the Trump administration", launched a almost unprecedented attack on the President in the op-ed, saying Trump's "erratic" and "amoral" behaviour is "detrimental" to America's health.
Trump had some unusual idea about the national debt: Another anecdote involving Cohn, the former director of the National Economic Council, claims that Trump suggested printing money to pay off the national debt: "Just run the presses - print money", the Woodward book quotes Trump as saying.
He said his book "is as carefully done as you can do an excavation of the reality of what goes on".
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