The Royal Family are said to be "hurt" at the Duke and Duchess of Sussex's announcement they will step back as senior royals.
Prince Harry and Meghan did not consult any senior royal before making the statement, the BBC understands.
Buckingham Palace was "blindsided", said BBC royal correspondent Jonny Dymond, as talks about their future had begun but were in very early stages.
The Palace said there were "complicated issues" to work through.
In their statement on Wednesday, Harry and Meghan said they made the decision "after many months of reflection and internal discussions".
They said they intend "to step back as 'senior' members of the Royal Family, and work to become financially independent".
They plan to split their time between the UK and North America, while "continuing to honour our duty to the Queen, the Commonwealth, and our patronages".
"This geographic balance will enable us to raise our son with an appreciation for the royal tradition into which he was born, while also providing our family with the space to focus on the next chapter," the couple said.
The couple's announcement raises numerous questions over what their future royal roles will entail.
Former press secretary to the Queen Dickie Arbiter said exactly how they will execute their plan "remains to be seen".
He told Newsnight. "There's a question of security: who provides the security?
"Is it Scotland Yard, as it does provide royalty protection. Is it the Canadians that's going to be asked to provide it? Who is going to pay for it?"
Royal commentator Penny Junor said the couple's actions were reminiscent of Harry's mother, Diana, Princess of Wales. The suggestion that they had not consulted other royals before making their statement was "beyond bizarre".
She said: "It has echoes of Diana when she suddenly announced after her separation [from Charles] that she was stepping back from 50 of her charities without consulting anybody.
"The problem is that they are not working for themselves, they are working for a family firm and to be making announcements of this sort without consulting is beyond bizarre."
Graham Smith, a spokesman for Republic, which campaigns for an elected head of state, said Harry and Meghan's decision "raises questions about the monarchy's future" and will prompt taxpayers to ask how the couple's extra security and overseas lifestyle will be funded.
Mr Smith questioned why they were announcing they were working towards being financially independent, saying: "To suggest that they're not already financially independent is incredibly crass and belies a sense of self-entitlement and a lack of self-awareness that is common among royals."
He added: "This really is wanting to have your cake and eat it. They have said they will dip in and out of royal duties
Signs the couple were unhappy with their royal life have been apparent for some months.
In October last year, the Duchess of Sussex began legal action against the Mail on Sunday over a claim that it unlawfully published one of her private letters. The paper stands by its story.
At the time, Prince Harry said his wife had "become one of the latest victims of a British tabloid press" after a "ruthless campaign" of "relentless propaganda".
"Though we have continued to put on a brave face... I cannot begin to describe how painful it has been," he said.
In an ITV documentary last year, Meghan, who was born in the US, described motherhood as a "struggle" due to intense interest from newspapers. "Not many people have asked if I'm OK," she said.
On Wednesday the royal couple announced they would be adopting a "revised media approach" from the spring.
As part of the new strategy, published on their website, they will "engage with grassroots media organisations and young, up-and-coming journalists".
They will also pull out of the so-called royal rota system, where journalists and media representatives are given exclusive access to cover royal engagements on the understanding they share the material they gather.
"The current system predates the dramatic transformation of news reporting in the digital age," the couple said.
Harry is sixth in line to the throne - behind Prince Charles, Prince William and his three children.
A staging post to leaving the Royal Family?
Analysis by Jonny Dymond - BBC Royal Correspondent
This is clearly a major rift between Harry and Meghan on one part, and the rest of the Royal Family on the other.
There are far more questions than answers; what will their new role be? Where will they live, and who will pay for it? What relationship will they have with the rest of the Royal Family?
And there's the institutional question. What does this mean for the Royal Family?
It comes just months after Prince Andrew stepped back from his duties after a BBC interview about his ties to sex offender Jeffrey Epstein.
Some might see this as the slimmed-down monarchy that the 21st Century needs.
But Harry and Meghan reached people that other royals didn't. They were part of the reinvention and refreshing of the institution.
We're now in wait-and-see mode as to whether this new model of being a royal can work - or if this is really a staging post for them to leave the Royal Family.