How can I recover a game center account where I forgot ID and password?

A long time ago, I made a game center account to which I attached my CoC village. This account hasn't been used anymore, and I have forgotten the ID and the password.

Now I want to transfer that game to my new phone which has a new game center ID, but need a transfer code from the old account to do so. Is there a way to recover my old game center account details?