The Uruguayan hit-man was strongly linked with a move away from Anfield over the summer, as he openly chased a transfer to the La Liga giants.
However, the Merseysiders held firm to keep their talismanic forward.
Suarez has since apologised for his actions and made a goal-scoring return to the first-team with a two strikes in two games following a 10-match ban for biting Chelsea’s Branislav Ivanovic last season.
But, he is believed to remain desperate to leave Liverpool in search of Champions League football.
Despite having invested almost £150m in new players over the summer, Real are ready to make a move for Suarez.
Liverpool rebuffed numerous offers in the region of £40m from Arsenal for their number seven before the transfer window closed, but are thought to be willing to accept £42m from a foreign side.
Real chief Carlo Ancelotti is keen to add a star centre-forward to his ranks to complete a fearsome attacking trio with the two most expensive players in the world – Gareth Bale and Cristiano Ronaldo.
Suarez’s representatives are believed to be holding talks with the Spanish club.
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