Earl Spencer has alleged in his forthcoming memoir that the then-Prince Charles said Diana, Princess of Wales, would be forgotten shortly after her death, during a heated dispute over arrangements for her 1997 funeral.
Buckingham Palace has strongly challenged the account, questioning Spencer's recollection of events nearly three decades after his sister's death.
In Swan Song: Diana, My Sister, Spencer claims he objected after being told by a senior Palace aide that Diana's sons, Prince William, then 15, and Prince Harry, then 12, would walk alongside their mother's coffin.
Spencer said he believed Diana would not have wanted her sons to participate in the funeral procession and insisted that it should not happen.
According to extracts published by the Daily Mail, Spencer then received a telephone call from Charles, who argued that the princes should take part.
Spencer alleges that during the ensuing argument Charles told him: "Rest assured, we'll forget her soon enough."
Spencer writes that he was stunned by the alleged remark and ended the call after responding that he could not believe what had been said.
The memoir is expected to be highly critical of the Royal Family and is due to be published worldwide next Tuesday.
BBC News said it had not seen the book in full.
Buckingham Palace responded by saying that it does not generally comment on books but suggested grief could affect recollections.
A spokesperson said the King was mindful that "the pain of fraternal grief can cloud reason, affect judgment and colour memory" many years after a loss.
Prince Harry has previously discussed his experience of walking behind Diana's coffin at age 12.
He said in 2017 that no child should be asked to do so, while also saying he was glad, looking back, to have participated in the day.
The memoir's release has also been accompanied by reports that four copies of its unreleased Dutch-language edition were stolen from a truck in the Netherlands while being transported from printers to the publisher's warehouse.