The Egypt international was sensational under Jurgen Klopp this season, with the winger netting 44 goals in all competitions.
Salah, 25, has been linked with a move to the Bernabeu in recent months, with Real president Florentino Perez desperate to sign a new Galactico.
And according to Spanish news outlet Don Balon, Real have opened talks with Salah’s representatives over a summer move.
They say Real are preparing a huge offer for the Liverpool superstar.
However, Real’s move for Salah depends on Gareth Bale.
Real will only move for the Egypt winger if Bale decides to leave the Bernabeu this summer.
The Wales international put clubs on alert after the Champions League final, with the attacker claiming he may leave to get regular game time.
Speaking after Real’s 3-1 win against Liverpool in the Champions League final on Saturday night, Bale, who netted twice, said: “I feel I should be playing week-in and week-out and I haven’t been.
“I haven’t been playing as much as I would like and I’m going to have to sit down in the summer and seriously think about what move I’m going to make next.
“I feel like there’s still a lot more to give for me and that’s why I need to be playing week-in and week-out and that’s what I need to be doing.”
Salah was forced off during the Champions League final with a shoulder injury after a clash with Real star Sergio Ramos.
It’s unclear whether the winger will be fit for the World Cup in Russia next month.
Salah will travel to Spain today to undergo treatment on his shoulder.
“Mohamed Salah began his treatment programme on Sunday, when he confirmed in a telephone conversation with the technical and administrative staff of the team he was in high spirits and determined to return quickly to the ranks of the team,” the Egyptian Football Federation said in a statement on its website.