Meghan, then 33, met McIlroy while staying with a close friend near the Fitzpatrick Manhattan Hotel in New York.
During the ice bucket challenge craze of 2014, McIlroy "unexpectedly" nominated Meghan.
In 'Revenge: Meghan, Harry and the war between the Windsors', Mr Bower writes: “She accepted the challenge, on condition that Rory came to her friend’s apartment and personally poured the bucket over her on the balcony.
"After the video was posted, the two went to the Fitzpatrick hotel for a drink.”
Meghan Markle with Rory McIlroy in 2014 Twitter/Rory McIlroy
Meghan Markle married Prince Harry in 2018 PA Features Archive/Press Association Images
He continues: “The following morning, McIlroy arrived at the Ringwood golf course to play in a new competition.
"Worse for wear after a hectic night, he fell back to 101st place.
"His performance suffered. ‘I wasn’t quite on my game,’ he confessed. ‘I was enjoying myself’.”
Eight weeks later, Meghan travelled to Dublin, Ireland, for a conference.
Mr Bower adds that she had dinner with McIlroy on the night of October 17.
The investigative journalist also writes about how Meghan was asked to appear on the cover of Vanity Fair's September 2017 issue.
The magazine ran with the headline 'Wild About Harry' on its front page – choosing to focus on the former Suits star's relationship with the prince, rather than her charity work.
When pre-publication copies were sent to the Palace, Meghan is said to have reacted "hysterically" as she "furiously" phoned her PR firm.
In the 2017 culture magazine, Meghan said: "I'm sure there will be a time when we will have to come forward and present ourselves and have stories to tell, but I hope what people will understand is that this is our time.
"This is for us. It's part of what makes it so special, that it's just ours. But we're happy.
"Personally, I love a great love story."
Mr Bower reports that Meghan was livid when Vanity Fair chose to lead their feature on these quotes, rather than focus on their philanthropy.