Reports in recent weeks have suggested that Gunners manager Arsene Wenger has sanctioned the sale of the Egypt international after he failed to force his way into the first-team picture during his first 18 months with the club.
Elneny impressed during his side's penalty-shootout win over Chelsea in the Community Shield on Sunday, however, and told reporters afterwards that he is determined to stay and fight for his place.
"I am staying at Arsenal this season," the 25-year-old is quoted as saying by The Sun. "There has been a lot of criticism recently, and I wanted to prove that I deserve to play for a big team like Arsenal.
"That was my best game for them.
"Winning the Community Shield gives us momentum ahead of the new season. We players talked about the importance of winning the game."
Since joining from Basel in January, Elneny has made 39 appearances for Arsenal in all competitions.