The Mirror says Salah, 29, was named in his country's preliminary squad for the summer Games as one of the three overage players permitted in the Under-23 squad, with Arsenal midfielder Mohamed Elneny also included.
But the Reds are yet to sanction their player travelling to Japan for the Olympics.
Salah taking part for Egypt would allow him almost no time to prepare for the start of the Premier League season in August.
"His position isn't clear yet, we will try one last time with Liverpool to let him participate," Egyptian FA chief Ahmed Megahed told ONTime Sports on Sunday.
"Mohamed wants to be in the Olympics, and we are working to make it happen."