Crisis meetings involving senior royals have taken place following the Duke and Duchess of Sussex's interview with Oprah Winfrey.
BBC royal correspondent Daniela Relph said Buckingham Palace "will not want to feel rushed into saying something" about Prince Harry and Meghan's claims.
The couple spoke about racism, mental health, the media and other members of the Royal Family in the interview.
Meghan's father Thomas Markle has criticised the interview's timing.
The duchess - who is the first mixed-race member of the modern Royal Family - said a low point came when Harry was asked by an unnamed royal family member "how dark" their son Archie's skin might be.
Prince Harry later clarified to Oprah that the comments were not made by either the Queen or the Duke of Edinburgh.
Meanwhile, US President Joe Biden's spokeswoman said he would praise anyone for having the courage to speak out about mental health.
White House press secretary Jen Psaki described Prince Harry, 36, and Meghan, 39, as "private citizens" who were "sharing their own story in their own struggles".
An average of 11.1m people in the UK watched the two-hour interview when it was screened on ITV on Monday night after being broadcast in the US by CBS the previous day.
In the interview, the duchess said that she found royal life so difficult that at times she "didn't want to be alive anymore", and when she approached the institution for help, she did not get it.
Labour leader Sir Keir Starmer said that Meghan's allegations about racism and a lack of mental health support should be taken "very seriously".
Prime Minister Boris Johnson declined to comment on Monday, but said he has "always had the highest admiration for the Queen and the unifying role she plays".
He said "when it comes to matters to do with the Royal Family, the right thing for prime ministers to say is nothing", after being asked specifically if he believed the Royal Family was racist.
During the interview, the duchess was asked why she thought the Royal Family did not make Archie a prince - which Meghan said she wanted so that he would get police protection.
"In those months when I was pregnant, all around this same time so we have in tandem the conversation of he won't be given security, he's not going to be given a title, and also concerns and conversations about how dark his skin might be when he was born," Meghan said.
She said the remarks about skin colour were made to Harry and he relayed them to her.
The couple's children do not automatically become princes or princesses due to a rule that has been in place since 1917 - unless the Queen steps in.
'Damaging'
Asked by Oprah whether there were concerns that her child would be "too brown" and that would be a problem, Meghan said: "If that is the assumption you are making, that is a pretty safe one."
When pressed, she refused to reveal who the individual was, saying: "I think that would be very damaging to them."
Prince Harry also refused to give further details, saying: "That conversation, I am never going to share."
Prince Harry said that none of his relatives spoke out in support of Meghan about the "colonial undertones" of news headlines and articles.
"No-one from my family ever said anything over those three years. That hurts," he said.
The couple moved to California after formally stepping down from royal duties in March 2020, and it was announced last month that they would not be returning as working members of the Royal Family.
In footage not included in the original interview, Prince Harry said racism from the tabloid press that filtered into the rest of society was a "large part" of why he and his wife left the UK.
The Society of Editors said it was "not acceptable" to claim sections of the press were bigoted "without providing any supporting evidence".
Labour MP Diane Abbott said she could cite "story after story when Meghan was treated quite differently from white members of the Royal Family".
Ms Abbott described the impact of coverage she has received as "hurtful", adding: "It's corrosive and in the end, it makes you doubt yourself."
In other key revelations from the Oprah interview:
- The couple announced their second child, which is due in the summer, is a girl
- They exchanged vows in a ceremony led by the Archbishop of Canterbury in their "backyard" three days before they were legally married at their public wedding in May 2018
- Prince Harry said his brother and father were "trapped within the system" of the Royal Family
- He said his family cut him off financially at the beginning of last year and his father stopped taking his calls
- But the prince said he loved his brother "to bits" and wanted to heal his relationship with both him and his father
- Meghan said she phoned the Queen after Prince Philip went into hospital last month
In further comments, Meghan told Oprah she felt betrayed by her father Thomas Markle's conduct in the run up to her 2018 wedding with Prince Harry.
Mr Markle told ITV's Good Morning Britain on Tuesday that the interview was the first time he had heard his daughter speak for several years.
"I'm very disappointed about it. I've apologized about this thing, what happened, at least 100 times or so," he said.
"At the time it was awkward, I was a bit shocked," he added.
He described the interview as "over the top" and ill-timed, but said he did not believe the Royal Family was racist and that the comments about Archie's skin colour may have simply been a "dumb question".
Asked about Meghan's claim she had "lost" her father, Mr Markle said: "I'm available any time we can get together. I'd love to get together. I'd certainly like to see my grandson."
He also criticised the support from the Palace when he and his family were under intense scrutiny from the media, saying: "Nobody was there to care for us, nobody looked after us."
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