The Reds have entered the race to sign the highly-rated attacker with Napoli and Fiorentina.
Callejón’s career at the Bernabeu has been very stop-start, having moved to Espanyol for three years in 2008, before returning to the Spanish capital.
The Daily Express reports:
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.
It could be all change at Real this summer, with manager Jose Mourinho having left to return to Chelsea, being replaced by Carlo Ancelotti.
The Spanish giants have also been linked with a move for Reds forward Luis Suarez for some time.
The Shakhtar Donetsk midfielder’s move to Anfield has been on the cards for some time and reports this weekend suggest Liverpool have finally agreed a £25million fee for his services.
Brendan Rodgers has already added defender Kolo Toure, attackers Luis Alberto and Iago Aspas and goalkeeper Simon Mignolet to his squad this summer.
And finally, two Reds are in action tonight in the FIFA U20 World Cup.
Conor Coady will captain the England U20 team in their game against Egypt U20, whilst Jon Flanagan will also start for the Lions.
England U20 vs Egypt U20 gets underway at 7pm (UK) and is live on British Eurosport.