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 to Oprah about racism, mental health, the media and other members of the Royal Family.
Prince Charles did not speak about the interview during a visit in London.
An average of 11.1m people in the UK watched the interview on Monday night.
In it, 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".
Former US secretary of state Hillary Clinton also backed Meghan, saying: "This young woman was not about to keep her head down, you know, this is 2021."
The two-hour interview was screened on ITV on Monday night attracting 12.4m viewers at its peak - the broadcaster's biggest audience since the 2019 Rugby World Cup final - while it was viewed by an average of 17.1m in the US 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 any more", 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".
No 10 said Prime Minister Boris Johnson watched the interview on Monday night, but declined to comment further.
It followed Mr Johnson's comments earlier on Monday, when he said he had "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.
Meanwhile, a YouGov opinion poll of 4,656 people conducted after the interview aired in the UK suggested:
- The public was equally split 32 per cent by 32 per cent on whether Prince Harry and Meghan were treated fairly or unfairly by the Royal Family - while 36 per cent said they did not know
- Among over-65s, 50 per cent thought the couple were treated fairly, compared with just 13 per cent of 18 to 24-year-olds
- There was little to no change in people's opinions when compared with an identical poll carried out last week
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.
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.
"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.
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.
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 apologised about this thing, what happened, at least 100 times or so," he said.