The Champions League winners were heavily linked with the Uruguayan last year, in an episode that almost saw him head away from Anfield.
Suarez actively chased a switch, citing a lack of top level European football as a major factor in his call, with Real his desired destination.
In the end the Spaniards snapped up Gareth Bale in a world record deal instead, leaving the Kop hero angling at a move to Arsenal.
But Liverpool held firm to keep their man, who has since committed his future to the Merseysiders by penning a mega money contract and declaring his happiness at his current location.
But that has not deterred Real, who are keen to line him up alongside Bale and Cristiano Ronaldo.
Despite their lavish spending in recent years, Los Blancos are prepared to offer £76m in the hope that Brendan Rodgers’ side may relent.
Real are also willing to throw in Fabio Coentrao or Alvaro Morata – both of whom have been targets for Liverpool – to tempt them to sell.