Jaziri spent last season on loan at Zamalek from Arab Contractors before the Cairo club made the move permanent this summer.
His destiny was unclear though, given that Zamalek were banned from two transfer windows and fined $200,000 after striking a new deal with youngster Hossam Ashraf while he had already signed for a Cameroonian academy.
"FIFA notified Zamalek that we can register Jaziri in the club's squad for next season," Zamalek's board member Amr Adham told Egyptian football website FilGoal.com.
There was also uncertainty over Ounajem's fate, with the winger having spent last season on loan at Morocco's Wydad before returning to Zamalek in the close season.
Zamalek's lawyer Nassr Azzam confirmed that Ounajem has been cleared to play with the club, subject to the Egyptian FA receiving his international clearance from the Moroccan Football Federation, which is largely a formality.
The club had appealed FIFA's ban but should it stand, Zamalek will not be able to register their three new signings: winger Omar Kamal, defender Mohamed Shabana and Tunisian right-back Hamdi Nagguez.
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