Cenk Tosun joined Everton from Besiktas over the winter transfer window.
The Turkey international has been dropped for the last two games in favor of Oumar Niasse.
Tosun did not start in the 5-1 defeat to Arsenal but did make an appearance off the bench.
Reports in the Turkish media have claimed that Tosun is growing frustrated with his lack of opportunities.
There are even rumors that he is considering returning to his former club Besiktas.
The 26-year-old has denied even considering leading Everton until he has proven himself.
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“I’ve zero interest going back to Turkey at this point,” he added.
“Yes I’m a Besiktas fan, but I’ll never return to them without fighting and achieving things in Everton. That’s not me.
“I’ll show everyone in England who is Cenk Tosun.
“I have a dream. I never quit a task before completing it.
“After I complete my mission in here, and if the conditions are suitable I’d love to go back to Turkey but it’s too early for that.
“I’ve just started to know my team-mates. In Besiktas, my team-mates knew where I was going be on the pitch before they passed. They knew what I was going to do. In here they’ll learn that too in time.”
Tosun joined Everton having scored 14 times and provided three assists in all competitions.
The Turkey international was on fire in the Champions League and was directly involved in six of the Black Eagles 11 goals in the group stage.
The Istanbul giants ended up finishing Group G as leaders and will take on Bayern Munich next in the Last 16.
Tosun meanwhile will face Crystal Palace on Saturday, if of course, Allardyce decides to give him the nod over Niasse who has not scored in his past two games.