During a post-presidency visit to the billionaire's private island in the British Virgin Isles, the former First Lady purportedly said: "We're free."
Ms Obama asked staff on the island to call her Michelle, the Virgin founder claimed inexcerpts of the book, first published in America'sPeoplemagazine.
"It's so nice to have my name back after eight years," she said.
Barack Obama, a keen surfer, did not have the chance to do watersports during his two terms in office, according to the Virgin Group boss.
But on his first holiday following his presidency, he learned to kitesurf.
Branson wrote that at the end of their last session, Mr Obama "was standing shirtless at the back of the boat with an invigorated look on his face, muscles bulging and eyes beaming".
Barack Obama's best moments in the White House
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He added: "I looked at him and thought: ‘Eat your heart out, Putin!'”
Mr Obama had a optimistic view towards the coming years, Mr Branson wrote, even as Donald Trump vowed to dismantle his predecessor's legacy.
“He told me Michelle was more of a glass half-empty attitude, whereas he is a more glass half-full type of person."
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