Sources close to Al Ahly told ESPN that Liverpool have been monitoring Hegazi and have explored the possibility of signing the defender.
The 26-year-old joined West Brom on loan in July and has been ever-present in the first team at the Hawthorns, starting in all 14 of the club's Premier League games so far this season.
ESPN said Leicester City are also keen on signing Hegazi, with the Foxes' head of player recruitment Eduardo Macia previously having worked with the defender at Fiorentina.
However, West Brom are reported to have an option in Hegazi's loan deal to sign him on a permanent basis for €5m (£4.4m).
Liverpool have been urged to bring in defensive reinforcements in the January transfer window to enhance their chances of finishing in the top four.
Former Liverpool defender Phil Thompson last month urged the club to invest in the January transfer window to solve their defensive issues.
"I do feel that come January we need to improve the defenders and get an out-and-out striker," Thompson told Sky Sports.
"[Liverpool manager Jurgen Klopp] has brought players in for the defensive positions but while they're good players, none of them are great players who you'd say would make it into the other top five sides.
"I don't believe the Virgil van Dijk signing alone would have solved our problems. He'd improve our defence but I don't think he's the sort of vocal leader who would sort it out. I'd like to see a more physical and commanding presence in the side," Thompson added.
Sixth-placed Liverpool face Stoke City at the Britannia Stadium on 29 November.