The Daily Mail says Real will trigger the latest saga in Suarez's colourful career by bidding around £76million for the Liverpool striker.
The player himself has said that there is nothing to be concerned about because he is happy on Merseyside and the club is going in the right direction.
However, even those close to the 27-year-old admit they can't say for sure what his final answer will be if Madrid keep increasing the mark on their bid.
Real will also offer various makeweights with Fabio Coentrao, who has agreed a new contract, and Alvaro Morata already mooted.
Suarez is currently with the Uraguay squad ahead of the World Cup. He is expected to recover from injury in time to take his place when Uraguay meet England in 3 weeks time.