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 it was “increasingly unsustainable for Buckingham Palace to say nothing” about Prince Harry and Meghan’s claims.
She said the Palace “will not want to feel rushed into saying something”.
Prince Harry and Meghan spoke about racism, mental health, the media and other royals in the interview.
Meghan – 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.
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.
‘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 new footage not included in the original interview, Prince Harry was asked if the couple left the UK because of racism, and replied: “It was a large part of it.”