His comments came after Cardinal Vincent Nichols, the head of the Catholic Church in England and Wales, suggested a coming together for Philip’s funeral could help heal any tensions.
The Duke of Sussex laid bare a rift with members of his family during an interview with Oprah Winfrey last month.
Asked during an appearance on the BBC’s Andrew Marr programme if he agreed with Mr Nichols, Sir John said: “I’m sure he is right, I hope he’s right, I believe he is right and I certainly hope so.
“The friction that we are told has arisen is a friction better ended as speedily as possible, and a shared emotion, a shared grief, at the present time because of the death of their father, their grandfather, I think is an ideal opportunity.
“I hope very much that it is possible to mend any rifts that may exist.”
Speaking on Times Radio on Saturday, Mr Nichols said “many a family gather and get over tension and broken relationships at the time of a funeral”.
“Something very profound unites them all again. And that would be true for this family, I’m sure,” he said.
“Obviously the whole ceremony will be watched by everybody, but you think of the complexities of the dynamics in that family and we have to think of Harry so far away. I’m sure he’ll come but not being,…