The Spanish giants tracked the Reds’ talismanic forward last summer, but ultimately elected not to mount a serious offer despite the player himself indicating that he was interested in making the move.
Liverpool stuck by their man despite after a series of controversial incidents and his public transfer saga during 2013, and rewarded him for his fine start to the season with a new £200,000-per-week contract before Christmas.
Even though this deal contains clauses that have put his value up to around £70m, Real remain keen.
However, much depends on Liverpool’s final finishing position this season.
If the Merseysiders manager to qualify for the Champions League, Suarez may elect to stay after recent rumours have indicated that he is settled at the club after a turbulent spell.
Real could push for a transfer if Brendan Rodgers’ men cannot match their number seven’s ambitions.
The Bernabeu side are also interested in Arsenal target Julian Draxler.
The German ace is seen as an ideal alternative to Suarez.