When the documentary-maker pressed him again, he replied: "I think it's a very tough job being prime minister.
"Obviously, if the ball came loose from the back of a scrum - which it won't - it would be a great, great thing to have a crack at. But it's not going to happen."
Mr Johnson's sister Rachel said the mayor, who at the age of 12 painted a self-portrait resembling a Grecian demi-god according to the documentary-maker, set his sights high when he was a child.
"As Boris was growing up whenever anyone asked him what he wanted to be, he would answer - 'World King'," she said.
She said of her brother's current relationship with the Prime Minister, with whom he went to school at Eton: "When they're together it's rather sweet, because David Cameron - even though he's taller - looks at Boris as if he's still head boy at Eton. Remember, Cameron was two years younger - the young pup."