The arrival of Alonso on a three-year contract follows the earlier signings of Martin Petrov and Robbie Blake on free transfers.
The highly rated 19-year-old, who has represented Spain at under-19 level, made his Real Madrid debut last season but found further first-team opportunities scarce at the Santiago Bernabeu.
Alonso comes from a rich footballing background, as both his grandfather and father played in La Liga for Real Madrid and Barcelona respectively.
Manager Owen Coyle admitted his delight at getting a player who he has been scouting for a long period of time.
"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," 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."
Alonso joins Jlloyd Samuel and Paul Robinson in competing for the left-back spot at the club.