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 Winfrey 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".
The two-hour interview was broadcast on Sunday by CBS in the US and was screened in the UK on ITV on Monday night.
In their interview, the duchess said that she found royal life so difficult that at times she "didn't want to be alive any more", and when she approached the institution for help, she did not get it.
Labour's shadow education secretary Kate Green said Meghan's claims were "really distressing, shocking".
She told Sky News: "If there are allegations of racism, I would expect them to be treated by the palace with the utmost seriousness and fully investigated."
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.
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."
"At the time it was awkward, I was a bit shocked," he added.
Oprah later revealed that the prince had shared with her that the remarks had not come from his grandparents - the Queen and Duke of Edinburgh.
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."
In other key revelations:
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