The PIF source said the wealthy Gulf monarchy wanted to sign the Egyptian winger in January during the next transfer window to join stars such as Cristiano Ronaldo in Saudi Arabia.
PIF holds a 75 per cent share in Al-Hilal, Al-Nassr, Al-Ahli and Al-Ittihad, but the source said it was not alone in wanting the Arab world's biggest football star.
"There is a competition inside the Saudi league who will bring Salah," the source said adding that a club affiliated with Saudi Arabia's state-owned oil and gas company was keen. "Aramco's Al Qadsiah has shown an interest too. So it's not only the PIF affiliated clubs"
Cristiano Ronaldo plays for Al-Nassr, Salah's former Liverpool team-mate Darwin Nunez is at Al-Hillal, another former Premier League player of the season N'Golo Kante is at Al-Ittihad but Salah is the biggest football star from an Arab country.