Other eye-catching moments in the book include Trump allegedly saying in one meeting: "My f****** generals are a bunch of p******.
"They care more about their alliances than they do about trade deals."
Trump also reportedly called Kim Jong Un, the North Korean leader, "beyond smart" and said Kim gave a graphic account of having his own uncle killed.
The comments appear in Rage, by Woodward - one half of the journalistic double-act that blew open the Watergate scandal, which toppled President Richard Nixon in 1974.
Extracts of the book were reported by CNN and the Washington Post.
Some of the most politically damaging parts of the book for Trump are revelations of how early he was warned of the seriousness of the pandemic.
On January 28, Robert O'Brien, the national security adviser, reportedly told Trump the virus "will be the biggest national security threat you face in your presidency".
On February 7, Trump, in one of 18 interviews he gave Woodward, stressed the seriousness of the virus. The audio was released today. "You just breathe the air and that's how it's passed," Trump said of Covid-19. "And so that's a very tricky one. That's a very delicate one. It's also more deadly than even your strenuous flu."
He added for emphasis: "This is deadly stuff."
The comments have led to accusations that Trump misled the American people about the seriousness of the virus, given at the time he repeatedly compared Covid-19 to flu and said it would simply "go away".
It was not until March 13 that the Trump Administration declared a national emergency over the pandemic.
CORONAVIRUS: TRUMP'S PUBLIC VS PRIVATE COMMENTS
JAN. 22
"We have it totally under control." — During CNBC interview on sidelines of economic forum in Switzerland. A day earlier, federal officials reported the first case of Covid-19 in the United States. "It's one person, coming in from China," Trump said. "It's going to be just fine."
JAN. 24
"China has been working very hard to contain the Coronavirus. The United States greatly appreciates their efforts and transparency. It will all work out well. In particular, on behalf of the American People, I want to thank President Xi!" — Trump tweet.
JAN. 30
"Hopefully it won't be as bad as some people think it could be. But we're working very closely with them and with a lot of other people and a lot of other countries. And we think we have it very well under control." — During trade event in Michigan.
FEB. 7
"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." — Trump phone interview with Woodward who asked the President about a call with Xi.
FEB. 10
"I think the virus is going to be — it's going to be fine." — During New Hampshire rally.
FEB. 26
"The 15 (case count in the US) within a couple of days is going to be down to close to zero. ... This is a flu. This is like a flu." — During White House coronavirus task force briefing.
MARCH 6
"You have to be calm. It'll go away." — During visit to Atlanta headquarters of the Centres for Disease Control and Prevention.
MARCH 7
"No, I'm not concerned at all. No, I'm not. No, we've done a great job." — After working dinner with Brazilian President Jair Bolsonaro.
MARCH 13
"We've done a great job because we acted quickly. We acted early. And there's nothing we could have done that was better than closing our borders to highly infected areas." — During Rose Garden announcement declaring a national emergency.
MARCH 16
"I've spoken actually with my son. He says, 'How bad is this?' It's bad. It's bad. But we're going to — we're going to be, hopefully, a best case, not a worst case. And that's what we're working for." — During White House briefing.
MARCH 19
"To be honest with you, I wanted to always play it down. I still like playing it down, because I don't want to create a panic." — Interview with Woodward.
MARCH 30
"I want to keep the country calm. I don't want panic in the country. I could cause panic much better than even you." — Responding to reporter's suggestion that he offered false assurances to Americans.
SEPT. 9
"I love our country and I don't want people to be frightened. I don't want to create panic, as you say. 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." — Talking to reporters and dismissing Woodward's book as a "political hit job."
-AP
