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.
Prince Harry and Meghan spoke about racism, mental health, the media and other royals 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".
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 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".
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.
Source: BBC News
BDST: 1555 HRS, MAR 09, 2021