The 19-year-old has signed a three-year deal with the Trotters and becomes the club’s third signing of the summer.
“He is an exciting young player who has undoubted quality and he will bring that into our mix. I think that we have a real talent on our hands,” Bolton boss Owen Coyle told the club’s official website.
“I have said before that when we are investing money in transfers that I want to do it with young, talented players.
“He isn’t the finished article yet – he’s a young player making his way – but you don’t get through at Real Madrid without having a bit about you.”