According to Mr Morton, William was anxious about an upcoming photo call to mark his third year at Eton.
Harry had been held back a year at Ludgrove school and William was concerned the press would focus on his brother's "underachievement" in comparison to his own "steady academic progress".
"Diana promised to discuss the matter with his father [Prince Charles] the next day when she arrived home after a night in Paris with Dodi [Fayed].
"She never returned," Mr Morton writes.
Earlier this week, the Duke of Cambridge opened up about the pain he felt after Diana's death.
Speaking in a new BBC documentary on mental health, the father-of-three said: "I think when you are bereaved at a very young age, anytime really, but particularly at a young age - I can resonate closely to that - you feel a pain like no other pain."
Prince William went on to say British people have "a hard time expressing their emotions" and shared advice on broaching difficult subjects with friends and family.
Watch the video below for the full interview: