Queen Elizabeth reportedly wanted Sophie, Countess of Wessex to become a mentor to the Duchess of Sussex.
The late monarch - who died in September, aged 96 - did "everything that she could" to welcome Meghan into the royal family, according to Gyles Brandreth, who also revealed that the queen was thrilled to learn that Prince Harry and Meghan were naming their daughter Lilibet, after her paternal great-grandmother.
Gyles, who has written books about the royals, told the Daily Mail newspaper's Palace Confidential: "I do know that when she heard the fact they were abbreviating it often to Lili, she thought, 'That's rather nice.'"
Gyles also claimed that the queen and Prince Philip disapproved of Harry's tell-all interview with Oprah Winfrey.
He explained: "The queen, I know she expressed about the Oprah interview, I know the Duke of Edinburgh would've thought that a bad idea, simply on principle, don't talk about it. The queen, 'Television nonsense, I don't really worry about that.'"
Meanwhile, Netflix recently released the latest trailer for Meghan and Harry's new documentary series.
The trailer for 'Harry and Meghan' features a number of photos of the Duke and Duchess of Sussex as they describe their life together.
Harry suggests that he didn't want his wife to suffer a similar fate to his mother, Princess Diana, who died in a car crash in 1997.
Harry - who has Archie, three, and Lilibet, 18 months, with Meghan - says: "I didn't want history to repeat itself."
In the trailer, the 38-year-old prince - who stood down as a senior member of the royal family before relocating to the US with Meghan - says: "It's really hard looking back on it and thinking 'What on earth happened?'"
Harry also suggests that women who marry into the royal family feel a particularly intense pressure.
Over an image of senior royals on the Buckingham Palace balcony, Harry says: "There's a hierarchy of the family.
"The pain and suffering of women marrying into this institution, it's a feeding frenzy."