Tosun, who moved to Goodison Park from Besiktas in January for £19.5m, has only made two Premier League starts and one appearance from the bench in his time with the Merseyside club so far.
The Turkish club's vice-president told HaberTurk that Besiktas will welcome the striker back to his homeland if he wants to return in the future.
"I think he's happy now, but at Besiktas our doors are always open to him," Nur Cebi said. "I would love him to come back in the future. And I think at one stage of his career, he will want a return to Besiktas because we have a brilliant atmosphere here, and players want to play for this shirt again.
"As a board member I can confirm to you that we didn't want to lose Cenk. But he has targets, one of them is to play in the Premier League. After talks with Everton reached an advanced stage, we considered his wishes and made the deal with the English club."