Campbell joined the Gunners in 2011 and has spent three seasons on loan at Lorient, Real Betis and Olympiakos respectively.
|3/1||Arsenal are 3/1 with Bet365 to draw their opening Emirates Cup game with Benfica|
The 22-year-old impressed while at the Greek club last season, scoring eight league goals and notching in a 2-0 victory over Manchester United in the Champions League, before catching the eye at the World Cup with Costa Rica.
Campbell's performances have attracted interest from Galatasaray, but Aysal says Wenger will not part ways with the forward.
"Yes, I want him for our team," he said, according to Sporx. "But Arsene Wenger wants to keep him in the squad this season and benefit from him.
"Nothing's happening at the moment. Wenger won't allow this transfer."
Campbell could feature for Arsenal in the pre-season Emirates Cup clash with Benfica on Saturday afternoon.