Hillary Clinton had to eat humble pie twice on election night.

Democratic US Presidential nominee Clinton apologised to President Barack Obama for her bruising loss - right after calling Republican President-elect Donald Trump to offer him "congratulations," according to a new book chronicling her 2016 presidential bid called Shattered: Inside Hillary Clinton's Doomed Campaign.

"Mr. President, I'm sorry," she said, according to a Washington Post review of the book due out today.

The White House urged Clinton to concede as Trump claimed battleground states - some by slim margins - because Obama wanted to avoid a messy recount, according to news.com.au.

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"You need to concede," Obama told Clinton directly, later repeating the instruction to her campaign chairman John Podesta for good measure.

The directive came after Clinton ignored previous messages from White House staff to throw in the towel.

Around 7:45 on election night, when Clinton and her aides still thought they were headed to the White House, vote-counter Steve Schale told campaign officials they were going to lose the biggest battleground in the country, the Washington Post reported.

"You're going to come up short," Schale said.

In that moment Bill Clinton reportedly became furious. Hillary was stoic. Their cocksure aides started to blame one another.

According to the newspaper's review, the call with Schale marked the beginning of a riveting account of the final, dreadful hours of Clinton's long pursuit of the presidency, as told by reporters Jonathan Allen and Amie Parnes in their new book.

Donald Trump giving his acceptance speech after defeating Hillary Clinton. Photo/AP
Donald Trump giving his acceptance speech after defeating Hillary Clinton. Photo/AP

The "blow by blow" account of the drama on election night in the Peninsula Hotel in Manhattan reveals how Clinton's confidence at her victory was replaced by resignation as Trump won the biggest upset in modern political history on November 9 last year.

Clinton finally succumbed to the loss with the words: "Give me the phone", according to the book.

She reportedly then called Trump and said: "Congratulations, Donald.

I'll be supportive of the country's success and that means your success as president."

Former President Barack Obama. Photo/AP
Former President Barack Obama. Photo/AP

Minutes after the Democratic Presidential candidate made the call to Trump, and suppressed "the anger that touched every nerve in her body", she rang Obama.

She told him she knew she had "let her country down" and that his legacy lay "shattered at Donald Trump's feet".

"Mr President, I'm sorry," Clinton said.

Clinton conceded the loss publicly the next morning.