A source has said the Duchess feels she has not received a satisfactory response to her concerns.
The letters have emerged just weeks after Meghan confirmed her decision not to attend the coronation on May 6.
It is understood her letter was sent in reply to one from the King. He is believed to be the only senior member of the family to make contact with the Duchess after the interview.
A source claimed the letters make clear the identity of the senior member of the family who made the comment.
It is understood both the King and the Duchess acknowledged that the individual’s remark was not made with malice.
The Duchess is believed to have thanked the King for his words. The Duchess’s letter is also said to suggest that she had never intended to specifically accuse the individual involved of being a racist, but was raising concerns about unconscious bias.
One royal source suggested that while the exchange was warm in tone, it had not eased the tension between the two sides.
The Duchess is understood to feel that concerns she has raised, which crucially include the way in which bullying complaints against her were handled and the allegation that neglect by the institution led her to feel suicidal, have still not been resolved.
It was confirmed earlier this month that the Duke would attend the King’s coronation on May 6 alone, as the Duchess had opted to remain at home in California with their children.
The desire to put her own family first by staying with Prince Archie on his fourth birthday, which falls on the same day, was said to have been the deciding factor.
However, she may also have wanted to protect herself by sidestepping the inevitable criticism and attention that her appearance would have provoked.
Royal sources indicated they had never expected the Duchess to fly over for the occasion, aware that her private correspondence with the King had not been enough to prevent further mud-slinging from the Sussexes.
The continued souring of relations within the royal family has seen the Sussexes star in a six-part Netflix docuseries and the publication of the Duke’s memoir, Spare, in which he squarely took aim at his family.
While the issues raised during the racism row have never been resolved, the Duchess is said to have moved on and does not hold a grudge.
The royal race row involving Lady Susan Hussey, Elizabeth II’s longest-serving lady in waiting, last autumn has also not helped, although the Sussexes were said to acknowledge that the incident reflected outdated attitudes held by the older generation.
Lady Hussey swiftly resigned after it was claimed she had subjected Ngozi Fulani, a domestic violence campaigner, to an “interrogation” over her heritage at a Buckingham Palace event.
Prince William was quick to convey his own disappointment in his godmother via a spokesman, who said he agreed with “the course of action”.
The speed with which the Palace dealt with the furore appeared to prove the Sussexes’ allegations had made their mark.
The Palace is also aware of the Sussexes’ frustration that initial email correspondence about the coronation made no reference to their children and their potential involvement. The omission only fuelled their feeling that their family plays second fiddle to the Waleses.
Following the late Queen’s death, the Duke and Duchess confirmed they would like their children, Archie, 3, and Lilibet, 1, to use the Prince and Princess titles they had automatically inherited.
But the Palace’s delay in recognising the children’s elevated status caused frustration, not least as the Prince and Princess of Wales’ titles and those of their children were swiftly changed upon the late Queen’s death.
The Duke of Sussex made clear in television interviews in January that he was holding out for an apology from his family, when he insisted he wanted to reconcile with the King and the Prince of Wales.
Tensions with Charles were soothed when the King did eventually reach out to his son and the pair had a heart-to-heart, shortly before the Duke confirmed his attendance at the coronation.
But there were no such overtures to Meghan, with whom relations with the Prince and Princess of Wales remain at rock bottom.
The Duchess told Winfrey that “several conversations” had taken place between the unnamed member of the royal family and the Duke about what their unborn son would look like.
She told the US broadcaster it was “safe” to assume there were concerns about his skin tone. She declined to name the person involved, saying: “I think that would be very damaging to them.”
The Duchess did not use the word racist about the incident during the interview.
The Duke also refused to elaborate further but suggested it was just a single conversation that had taken place.
“It was awkward. I was a bit shocked,” he said.
At the time of the interview, the King, then the Prince of Wales, was said to have been “deeply concerned” about the racism allegations, which prompted crisis meetings at the highest level of the institution.
Sources suggested he felt “let down” by his son and daughter-in-law, who he had supported “more than he would care to say”.
In January, the Duke attempted to row back from the damaging allegations made in their interview with Winfrey, insisting he did not believe the royal family was racist and that he had never suggested they were.
The claim about the comment about Prince Archie was not repeated in his memoir.
The Duke told ITV’s Tom Bradby that when he and the Duchess revealed that a member of the family had raised concerns about the colour of their unborn son’s skin, they were not accusing them of racism.
Asked if he would not consider that “essentially racist”, he replied: “I wouldn’t, not having lived within that family.”
The comment was considered an olive branch from the Duke to his family and an apparent about-turn after the Palace was engulfed in a race crisis in the wake of his claims.
The Sussexes’ disclosure about the alleged incident caused shockwaves around the world and proved very damaging to the monarchy, prompting the late Queen to issue her response: “Recollections may vary.”
The King has long championed multicultural Britain and firmly believes it is his role to protect and celebrate such diversity.
He has recently given his backing to a research project into the royal family’s historical links to the slave trade, having previously expressed his “personal sorrow” over the suffering of transatlantic slaves.
It has raised the prospect that the King might even be prepared to pay reparations, although the Palace has insisted it is too early to discuss it.
The Duchess has expressed disappointment at the lack of support she received from the institution during her time as a working member of the family, when she felt she was subjected to a volley of racist abuse from which she was unable to protect herself.
The simmering resentment was only exacerbated following the death of Elizabeth II, when the Sussexes felt they were treated appallingly by the royal family.
A failure to personally inform the Duke that his grandmother had died before putting out a public statement was followed by an apparent about-turn over the Duke’s right to wear military uniform and an invitation sent in error to a state reception at Buckingham Palace.
Despite putting on a united front with the Prince and Princess of Wales when the foursome greeted well-wishers in Windsor, the tension behind the scenes was palpable.