The Egypt international has persistently requested a transfer to a bigger club throughout last season, which forced the technical staff to axe him on more than one occasion.
"I apologize for the club and the fans for all troubles I have caused previously because of my desire to go abroad," Zaki told Dream TV on Friday.
The former ENPPI forward revealed that he had met Zamalek chairman Mamdouh Abbas, discussing a contract extension with the Whites.
"I met with the chairman and I told him that I want a new deal at the club, to prove my loyalty.
"I will reject any other offers as well, because I want to stay with Zamalek," he added.
The 24-year-old striker joined Zamalek last season after an unsuccessful short spell with Russian side Lokomotiv Moscow.