The late monarch's lady-in-waiting has offered to meet the black charity founder who she is said to have questioned about her ethnicity, it is reported.
Lady Susan Hussey, 83, resigned yesterday as an honorary member of the royal household after Ngozi Fulani said she was repeatedly asked where she "really came from" during the Queen Consort Camilla's reception on violence against women.
Ms Fulani, who runs a black women's domestic abuse organisation, said that she was asked: "What part of Africa are you from?"
Lady Susan later apologised for "unacceptable and deeply regrettable comments," Buckingham Palace said in a statement.
She now wishes to meet Ms Fulani in person, and apologise to her directly.
Palace officials hope that a discussion between the two would lead to "lessons being learnt in a spirit of co-operation".
Ms Fulani has had a "direct discussion" with the Palace and they are "working out how to move forward".
ITV's Chris Ship said: "On this issue about whether Buckingham Palace has reached out to Ngozi Fulani, I hear a direct discussion has now taken place between both sides.
"They are now working out how to move forward after this week’s events."
The chief executive of Sistah Space, an East London refuge that provides specialist support for women of African and Caribbean heritage, first detailed her exchange with the royal household member in a Twitter post.
Ms Fulani said that when she told the woman she was from East London, she responded, "No, what part of Africa are you from?"
She also allegedly said: "I can see I am going to have a challenge getting you to say where you're from. When did you first come here?"
Ms Fulani said she later replied: "I am a British national, my parents came here in the 1950s," before Hussey responded with: "Oh, I knew we'd get there in the end, you're Caribbean."
The charity operator said she then corrected the aristocrat by responding with: "No, I am of African heritage, Caribbean descent, and British nationality."
Today, Ms Fulani told ITV's Good Morning Britain: "Although I didn't experience physical violence, what I feel I experienced was a form of abuse."
Describing how Lady Susan also touched her hair during the incident, she said: "I was stood next to two other women - black women - and she (Lady Susan) just made a beeline for me, and she took my locks and moved it out of the way so that she could see my name badge.
"That's a no-no. I wouldn't put my hands in someone's hair, and culturally it's not appropriate."
The Prime Minister on Thursday declined to comment directly on the row but said that it was right to "confront" racism.
Speaking to broadcasters in Downing Street, he said: "As I've talked about in the past, I have experienced racism in my life.
"But what I am pleased to say is some of the things that I experienced when I was a kid and a young person I don't think would happen today because our country has made incredible progress in tackling racism.
"But the job is never done. And that's why whenever we see it we must confront it.
"It's right that we continually learn the lessons and move to a better future."
Mr Sunak, who became leader of the Conservative Party in October following Liz Truss's resignation, is the UK's first Hindu Prime Minister and the first of Asian heritage.
He told broadcasters it would not be "right" for him to comment on the matter, but said: "As we've all seen, they've acknowledged what's happened and made an apology for it."
William, who is on a trip to Boston in the US with the Princess of Wales, backed the decision of his godmother to resign as a Lady of the Household.
A Kensington Palace spokesman issued a statement, saying: "Racism has no place in our society.
"The comments were unacceptable, and it is right that the individual has stepped aside with immediate effect."
The Palace moved swiftly to respond to Ms Fulani's tweets on Wednesday morning, saying it took the incident at Tuesday's reception "extremely seriously" and had investigated immediately.
It added, not naming Lady Susan, that the individual concerned had resigned and apologised and that the comments were "unacceptable and deeply regrettable".
The King, who acceded to the throne less than three months ago, and Camilla have been made aware of the situation, the Palace said.
