Saiz, 26, has penned a four-year deal at Elland Road after the Whites agreed an undisclosed fee with Spanish second-tier outfit SD Huesca.
Saiz only featured for Real’s B and C teams, and went on to play for Getafe and Atletico Madrid’s B teams before moving to Huesca.
He is Leeds’ eighth signing of the summer and joins the likes of Vurnon Anita, Madger Gomes and Mateusz Klich in moving to Yorkshire.
Stoke could face a battle to keep Xherdan Shaqiri as reports have claimed Mark Hughes wants him at Southampton next season.
Bayern Munich have entered the race to Emre Can, who continues to talk up a move away from Liverpool this summer.