The 65-year-old star and his civil partner David Furnish, 49, became parents on Christmas Day 2010 to a child conceived using a donor egg and born via surrogate, the Daily Mail reported.
John fears his child will struggle with bullies because he works in showbusiness and because he doesn't "have a mummy".
"Being the child of someone famous is a huge ball and chain around your ankles. It's going to be very difficult," the musician told Radio Times.
"At school other children will say, 'You don't have a mummy'. We've come a long way, but there's still homophobia and will be until a new generation of parents don't instill it in their children," he said, adding Zachary calls him 'daddy' and Furnish 'papa'.
John wants music to be a big part of his son's life but would not push him into showbiz.