I used to have my very own pet swan that was orphaned so we raised it from just a little cygnet, I have a photo somewhere of me as a toddler sat on its back, I recall no biting taking place. He was called George and he was awesome.
Geese on the other hand, they're vicious bastards, I know a chap who went for a wee in a bush as a child and one thought it would take a peck at his pecker, he's been phobic ever since funnily enough.