The Nigeria international has spent the season on-loan with Gala from Napoli and can be signed for a set fee of €75m.
La Repubblica says Napoli are set to receive that fee, but not from Gala.
Instead, Saudi Pro League giants Al-Hilal and big-spending UAE outfit Al Ain are both prepared to meet the clause.
Both clubs will compete in the summer's Club World Cup and want Osimhen registered in time for the tournament's June kickoff.
With Gala this season, Osimhen has scored 33 goals in 37 matches.