The Turkish news outlet claims that Hegazi, 28, has been made the number one target for the Istanbul giants, who are expecting Martin Skrtel to leave them when his contract expires in the summer.
It is understood that the Egypt international has been identified as a replacement for the former Liverpool defender.
Aksam adds that Hegazi is valued at £6.4million.
Good move for West Brom?
Hegazi may have been a bit of a liability for West Brom in the Premier League when they were relegated last season, but he has improved in the Championship.
The ex-Fiorentina centre-back has made 33 appearances across all competitions this season and the Baggies have failed to win any of the five Championship games that he has missed due to injury.
Although he did not play well against Leeds on Friday, Hegazi would be missed by West Brom if they fail to win promotion from the Championship this season.
However if they return to the Premier League, Darren Moore would be better off letting him go and trying to find an upgrade.
This is Futbol rating 5/10
Likely to happen?
Fenerbahce are the sort of team that would suit Hegazi.
The Turkish Super Lig is not a strong league but the Egypt international would be closer to home in Istanbul and might stand out a bit more playing for a team going for league titles.
Albion fans who like Hegazi should be concerned by this report.
However Moore is unlikely to make any decision until the end of the season.
This is Futbol rating 4/10