Callejón spent much of last season at another La Liga club, Espanyol, and was unable to establish a first team spot at Real Madrid.
The possibility of the Spanish club signing Gareth Bale from Tottenham would push him even further back in the pecking order, and therefore Callejón is seeking a move to ensure playing time.
Real Madrid's recent signing of Isco from Malaga could be the final straw, as it could result on Callejón not getting into the team at all.
However new Real Madrid boss Carlo Ancelotti is keen to give more chances to reserve team players as he seeks to change the club's playing style, and this could change Callejón's eagerness to move.