The Spanish club have made a surprise move for the 25-year-old England international who has also been the subject of bids from Aston Villa and Hull City.
United manager Louis van Gaal may be prepared to listen to offers for Cleverley, who has just 12 months left on his contract, as he looks to shake things up at Old Trafford following a poor start to the season.
United suffered Capital One Cup embarrassment at the hands of League One side MK Dons and have only taken one point from a possible six in the Premier League so far this season.
Van Gaal has already allowed Wilfried Zaha to rejoin Crystal Palace on a season-long loan deal and striker Danny Welbeck is also been linked with a move away from Old Trafford, with a number of Premier League teams interested.
Javier Hernandez is reported to be wanted by Juventus while Shinji Kagawa is attracting interest from a number of European clubs.