"No one in the royal family has talked to Meghan yet," Gayle King said

Prince Harry has spoken with his brother Prince William and father Prince Charles following Harry and Meghan Markle's interview with Oprah Winfrey.

Gayle King, who is friends with Oprah and the Duchess of Sussex, revealed on CBS This Morning on Tuesday that Harry spoke with William and Charles over the weekend.

"Well, I'm not trying to break news, but I did actually call them to see how they were feeling, and it's true, Harry has talked to his brother and he has talked to his father too," King said. "The word I was given was that those conversations were not productive. But they are glad that they have at least started a conversation.

"And I think what is still upsetting to them is the palace keep saying they want to work it out privately, but yet, they believe these false stories are coming out that are very disparaging against Meghan, still."

She added that "no one in the royal family has talked to Meghan yet, at this particular time."

"I think it's frustrating for them to see that it's a racial conversation about the royal family when all they wanted all along was for the royals to intervene and tell the press to stop with the unfair, inaccurate, false stories that definitely have a racial slant," King continued.

"And until you can acknowledge that, I think it's going to be hard to move forward. But they both want to move forward with this and they both want healing in this family. At the end of the day, that is Harry's family."

Commonwealth Day
Prince William, Kate Middleton, Prince Harry and Meghan Markle
| Credit: Phil Harris - WPA Pool/Getty

During the couple's sit-down with Oprah on March 7, Meghan said her son Archie was denied a royal title and royal protection — and that there were "concerns and conversations about how dark his skin might be when he's born."

Harry and Meghan did not identify who made the remarks, though Oprah said it was not his grandparents Queen Elizabeth and Prince Philip.

Last Friday, William hit back at the racism claims while on a royal outing with wife Kate Middleton, telling a reporter, "We are very much not a racist family."

When asked if he had spoken to Harry, he added, "No, I haven't spoken to him yet, but I will do."

During the sit-down, Prince Harry also said his father had stopped taking his calls for a period of time. "There's a lot to work through there. I feel really let down because he's been through something similar," he said. "He knows what pain is like. And Archie's his grandson."

He continued, "Of course, I will always love him, but there's a lot of hurt that's happened. And I will continue to make it one of my priorities to try and heal that relationship."

Prince Charles and Harry
Prince Charles and Prince Harry
| Credit: Samir Hussein/WireImage

King also addressed the recent bullying allegations against Meghan, saying, "The bullying thing was raised in 2018 and now there's an ongoing investigation about bullying from Meghan Markle, when anyone who has worked with her will tell you exactly who she is.

"You know, she's really a very sweet, caring person. And as I say, Meghan has documents to back up everything that she said on Oprah's interview. Everything."

Days before Harry and Meghan's Oprah interview, The Times in the U.K. reported that the Duchess of Sussex faced a bullying complaint made by one of her close advisers during her time as a working royal at Kensington Palace — allegations her office has strongly refuted.

The complaint, which The Times reports was made in October 2018 by the couple's former communications secretary Jason Knauf, said that Meghan drove two personal assistants out of the household and undermined the confidence of a third staff member.

"The Duchess is saddened by this latest attack on her character, particularly as someone who has been the target of bullying herself and is deeply committed to supporting those who have experienced pain and trauma," a spokesperson for the Duke and Duchess of Sussex said in a statement obtained by PEOPLE. "She is determined to continue her work building compassion around the world and will keep striving to set an example for doing what is right and doing what is good."