The 25-year-old passed a medical and signed a contract with an option for a permanent three-year deal which can be exercised before the end of June.
Faubert will be presented at the Bernabeu on Monday.
He is the Primera Liga side's third mid-season signing after they landed French midfielder Lassana Diarra and Dutch striker Klaas Jan Huntelaar.
Youth-team midfielder Dani Parejo has also been recalled from a loan spell and incorporated into the first team.
Faubert will help give coach Juande Ramos options on the right of midfield where they have lacked a natural wide player for the position.
WEST HAM FANS! Check FourFourTwo.com’s ever-expandinginterviews archivefor more stuff to read:
Web Exclusives Mark Noble, Les Ferdinand
One on One Les Ferdinand, Paolo di Canio, Gianfranco Zola
My Secret Vice Nolberto Solano, Martin Allen
Ask A Silly Question Sir Geoff Hurst
What Happened Next? Julian Dicks
Sing When You’re Winning Nick Frost
Plus! Mark Noble on Weekend Wonders and West Ham on Talentspotter
