He recalled on the programme that Harry, then four years old, teased William when they were in the car together travelling from London to Gloucester, an incident he said happened around 30 years ago.
He explained the brothers had been locked in an argument for most of the journey.
"I had the privilege of working with his mother, William and Harry and many other members of the royal family.
"I remember a moment of Harry as a four-year-old, travelling down to Gloucester one weekend.
“He and his brother had an argument in the back seat of the car being refereed by the nanny.
"And in the end, Harry said: 'You'll be king one day, I won't. Therefore I can do what I want!'
"Diana said 'Where the hell did he get that from?’”
The revelations come after Harry and Meghan's bombshell announcement last week that they want to step back as senior members of the royal family and become "financially independent".
Following the shock announcement, Queen Elizabeth called a royal summit, which involved Harry, William and their father Prince Charles, on Monday.
The Queen then issued a statement regarding the current state of play.
“Today my family had very constructive discussions on the future of my grandson and his family,” the statement read.
“My family and I are entirely supportive of Harry and Meghan’s desire to create a new life as a young family.
“Although we would have preferred them to remain full-time working Members of the Royal Family, we respect and understand their wish to live a more independent life as a family while remaining a valued part of my family."
The statement continued: “Harry and Meghan have made clear that they do not want to be reliant on public funds in their new lives.
“It has therefore been agreed that there will be a period of transition in which the Sussexes will spend time in Canada and the UK.
“These are complex matters for my family to resolve, and there is some more work to be done, but I have asked for final decisions to be reached in the coming days," she concluded.