The 23-year-old has signed a four-year contract after passing a medical, the club's official website said.
Spaniard Valero is the Baggies' ninth signing of the summer, taking their close-season spending past A#20million.
Manager Tony Mowbray said: "I'm delighted to welcome Borja to the club.
"From a footballing point of view he possesses many of the attributes I like in a player."
Mowbray said he believed Borja may need a little time to adjust to the Premier League, and that despite a record fee was not guaranteed a place in the team.
But he added: "Borja can link up the play from back to front and he will enhance the excellent midfield options we already possess."