Bob Woodward, who broke the Watergate scandal and is one of the nation’s most respected journalists, interviewed Mr Trump 18 times from December to July.
Mr Trump is quoted as telling him the virus was “deadly stuff” before the first US death was confirmed.
Responding, the president said he had wanted to avoid causing public panic.
Some 190,000 Americans have been recorded as dying with Covid-19 since the beginning of the pandemic.
On Wednesday, some US media released parts of the interviews between the president and the journalist, revealing his reported remarks on the outbreak as well as race and other issues.
This is President Trump on tape, on February 7, saying that the coronavirus is "more deadly than your – you know, your, even your strenuous flus." But he minimized the threat in public. On February 26, he told the public "I think that’s a problem that’s going to go away." pic.twitter.com/TOHTpqYtvZ
— Brian Stelter (@brianstelter) September 9, 2020
Here are some of the key quotes so far from Rage, which will be released on 15 September.
What does the book say about Mr Trump and the virus?
Mr Trump indicated that he knew more about the severity of the illness than he had said publicly.
“It goes through the air,” Mr Trump told the author on 7 February.
“That’s always tougher than the touch. You don’t have to touch things. Right? But the air, you just breathe the air and that’s how it’s passed.
“And so that’s a very tricky one. That’s a very delicate one. It’s also more deadly than even your strenuous flus.”
Speaking on Capitol Hill on 10 March, Mr Trump said: “Just stay calm. It will go away.”
Nine days later, days after the White House declared the pandemic a national emergency, the president told Woodward: “I wanted to always play it down. I still like playing it down, because I don’t want to create a panic.”
How did the White House react?
Speaking from the White House on Wednesday, Mr Trump told reporters: “I don’t want people to be frightened, I don’t want to create panic, as you say, and certainly I’m not going to drive this country or the world into a frenzy.
“We want to show confidence, we want to show strength.”
The president – who is running for re-election in November – said the Woodward book was “a political hit job”.
Responding to reporters’ questions on the book, White House press secretary Kayleigh McEnany said: “The president never downplayed the virus, once again. The president expressed calm. The president was serious about this.”
In a tweet, Mr Trump’s Democratic White House challenger Joe Biden said that “while a deadly disease ripped through our nation, [the president] failed to do his job – on purpose. It was a life or death betrayal of the American people.”
Leaders are responsible for keeping people calm, but there is a fine line between avoiding panic and making a crisis worse. President Trump told Woodward Covid-19 was deadlier than the flu, but in public he downplayed the danger.
Other leaders took a different approach. UK Prime Minister Boris Johnson said bluntly that people would die: “Many more families are going to lose loved ones before their time.”
Mr Johnson thought people should know the severity of the pandemic. In contrast, Mr Trump often downplayed the virus from the beginning. In recent weeks, his advisers began speaking of the coronavirus in the past tense – as if the problem were gone.
Scientists disagree, saying there is likely to be a surge in the autumn, following the pattern of other respiratory diseases. One fact is irrefutable, however. Mr Trump wants people to see him as a strong leader. He also wants them to go to the polls and vote – and not to worry about the virus.
What else did the book reveal?
Woodward writes that he brought up the Black Lives Matter protests in a conversation with the president on 19 June, suggesting that “white, privileged” people like themselves ought to work to understand how black Americans feel.
“You really drank the Kool-Aid, didn’t you?” Mr Trump said. “Just listen to you.”
The nationwide protests against police brutality and racism were sparked by George Floyd’s death in Minnesota in May.
Mr Trump also repeated the suggestion that he had done more for African Americans than any president aside from Abraham Lincoln, who abolished slavery.
Later, on 8 July, Mr Trump again reiterated that he had “done a tremendous amount for the black community”, but was “not feeling any love”.
The Washington Post also cited an interview where Woodward asked the president about whether America has systemic racism.
Mr Trump said while these problems exist everywhere, “I think probably less here than most places, or less here than many places”.
The president also acknowledged racism affects the lives of people in the US, saying “it’s unfortunate”.
Woodward’s book also cites dozens of letters between Mr Trump and North Korea’s Kim Jong-un. In the letters, filled with flowery language, Mr Kim reportedly referred to Mr Trump as “Your Excellency” and noted their “deep and special friendship will work as a magical force”.
According to US media, Mr Trump told Woodward of his connection with Mr Kim: “You meet a woman. In one second, you know whether or not it’s going to happen. It doesn’t take you 10 minutes and it doesn’t take you six weeks. It’s like, whoa. OK. You know? It takes somewhat less than a second.”
Mr Trump also reportedly told Woodward that he felt his predecessor, Barack Obama was “highly overrated”.
“I don’t think Obama’s smart,” Mr Trump reportedly said. “And I don’t think he’s a great speaker.”
According to CNN, Mr Trump told Woodward he made President George W Bush “look like a stupid moron, which he was”.