He subsequently made several television appearances in which he criticised the conduct of his daughter and the royal family.
According to Bower's book, the Prince "berated" the Duke and questioned why the Duchess could not just go to see her father.
The Duke told his father that his wife refused to phone Mr Markle because she suspected his phone "was (not) in his possession" and "his email account was compromised", the book said.
Bower writes: "Meghan's excuses irritated Charles and perhaps also the Queen.
"The monarch must have found it hard to believe that Meghan could not resolve her differences with Thomas Markle."
It is claimed that the Queen and the Prince joined the Duke and Duchess on a conference call on which they urged the Duchess to fly to America for "a reconciliation".
Bower writes: "Meghan rejected the suggestion...the conference call ended with both the senior royals perplexed."
Buckingham Palace declined to comment in line with its long-standing policy to never respond to unofficial books, as did Clarence House.
The Duchess later said during the same visit to Castle Mey "they fundamentally don't understand", referring to the palace, according to the book.
A spokeswoman for the Duke and Duchess of Sussex has not yet responded to a request for comment.
Bower's book claims to be a "characteristically explosive and rigorously researched account of the rift between the Sussexes and the rest of the Royal family, touching on every aspect of scandal and suspicion".
The biography, entitled Revenge: Meghan, Harry and the war between the Windsors, is out on July 21 and serialised by The Times, The Sunday Times and The Sun.