Salah has been in scintillating form since completing his £36.9m move from Roma during the summer, scoring 24 goals in just 29 matches across all competitions.
The 25-year-old's form has drawn interest from some of Europe's biggest clubs despite Liverpool's reluctance to lose the winger, and Egyptian FA president Hany Abo Rida has backed a move to Madrid.
"Real Madrid will make an offer for Salah during the summer," Rida is quoted as saying by the Liverpool Echo.
"Salah is now looking to improve his game. Obviously if an Egyptian plays for a club as popular around the world as Real Madrid, it's going to be great for Egyptian football."
Salah's national team coach Hector Cuper also suggested that the in-form winger would be a good signing for Madrid last month.