Granero, 24, has seen his opportunities limited in Spain this season and Anfield boss Kenny Dalglish is understood to be a keen admirer of his talents.
The player's agent, Teofilo Lazaro, stated: "He's finding his chances limited at Real. He's keen for a move away – possibly to Liverpool. If he can't play at Madrid then he'll play somewhere else. It's as simple as that."
Granero joined Madrid from Getafe in 2009.