The Hoops are looking to add some late reinforcements before the transfer window shuts on Friday, having already been linked with Real centre-back Ricardo Carvalho.
Granero, who has a year remaining on his contract in Madrid, is the latest target being linked with a move to Loftus Road.
The 25-year-old is already thought to be on the radar of Arsenal, Liverpool and Tottenham this summer but QPR could beat them to the punch after entering talks with Real over the Spaniard.
Granero came through the youth ranks with Real and moved to Getafe in 2008 before returning to his boyhood club a year later.
The Spanish midfielder made 24 appearances for Los Blancos last season.