The assertions, made in a soon-to-be published book Swan Song: Diana, My Sister, relate to the now-monarch's conduct in the days after Diana's death in 1997.
In it, the Earl of Spencer claims he had a heated conversation with King Charles, who at one point said "we'll forget her soon enough".
Buckingham Palace released a statement on Wednesday saying fraternal grief could colour recollections of events "even many years after such a loss".
Diana was killed at the age of 36 when the car carrying her and her lover Dodi al-Fayed crashed in Paris as it sped away from chasing paparazzi photographers on motorbikes.
Her brother, Charles Spencer, alleges the reported comments were made in an argument over whether Diana and Charles' children, princes William and Harry - then 15 and 12, should walk behind her coffin during her funeral.
The Earl claims he told Charles that Diana wouldn't have wanted her sons to go through something so traumatic, so publicly.
Charles then called the Earl, he claimed, and told him that he wanted his sons to participate in the procession, and cited his walking in the funeral procession of Lord Mountbatten 18 years earlier as an example of what he wanted his sons to follow.
However, when the Earl pointed out to Charles that he had been 30 at the time of Lord Mountbatten's death, he said the now king "went nuts".
The Earl writes in Swan Song: "Prince Charles continued his stream of anger, without me interrupting. Then he stopped.
"I assumed the pause signalled an effort to bring himself under control, but then he unleashed down the phone what I have always taken to be his pent-up contempt for Diana, and his horror that the world was so bowled over by her death.
"'Rest assured,' he hissed, 'we'll forget her soon enough!'"
The Earl was "so stunned" by Charles' alleged remarks, he wrote, that he took a few seconds before replying: "I can't believe you just said that".
Diana became the most photographed woman in the world after marrying Charles in a glittering ceremony in 1981. However, the marriage turned sour and the couple eventually divorced in 1996.
Asked about the report, a Buckingham Palace spokesperson said: "While we do not comment on books as a matter of principle, His Majesty is mindful that the pain of fraternal grief can cloud reason, affect judgement and colour memory in ways others do not recognise, even many years after such a loss."
Prior to the book's release, Spencer said: "The aim of this book is to tell the truth about Diana from her brother's perspective ... I want to speak on Diana's behalf and do so in an openly positive way."
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