Meghan said, when she was pregnant with Archie, an unnamed member of the royal family raised “concerns and conversations about how dark his skin might be when he’s born”.
The American duchess is the first mixed race person to marry a British senior royal in modern history.
A stunned Winfrey asked: “What? Who is having that conversation?”
Meghan paused and said there were “several conversations” with Harry about Archie’s skin tone, and “what that would mean or look like”.
Asked 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.”
Pushed by Winfrey on who had those conversations, Meghan refused to say, adding: “I think that would be very damaging to them.”
Harry refused to give further details, adding: “That conversation, I am never going to share. At the time it was awkward, I was a bit shocked.”
The duke said that none of his relatives spoke out in support of Meghan following the racism he said she faced in the media.
“No-one from my family ever said anything over those three years. That hurts,” Harry said.
“But I also am acutely aware of where my family stand, and how scared they are of the tabloids turning on them.”
Meghan spoke of her upset, particularly with the Commonwealth being such an important factor for the monarchy.
The duchess said: “I could never understand how it wouldn’t be seen as an added benefit and a reflection of the world today at all times, but especially right now, to go how inclusive is that you can see someone who looks like you in this family, much less one who’s born into it.”
The Sussexes have been vocal supporters of the Black Lives Matter movement.
Meghan previously delivered an impassioned Black Lives Matter speech to her old high school following the death of George Floyd in the US, sharing her “absolute devastation” at racial divisions.