The Uruguayan has stated that he wants to leave the Reds this summer after enduring a difficult end to last season.
Madrid are favourites to land the 26-year-old but Liverpool would like the Spanish side to make their intentions clear sooner rather than later.
A source from the Reds told The Mirror: "Everyone here remembers how Real behaved in 2004, when they came in with a very late bid for Michael Owen after a summer of constant speculation. They got the player for £7m, which was a fraction of his real value.
"The difference is that Michael only had one year left on his contact while Luis has three. This time we feel we are negotiating from a position of strength rather than weakness, and the £40m clause in Luis's contract would only give a club the right to speak to him."
Suarez is currently away on international duty with Uruguay at the Confederations Cup.