The Russian international joined the Gunners from Zenit St Petersburg for £15 million in 2009, but is set to depart once his contract expires, having made only 11 appearances for the club in all competitions last season.
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"Besiktas? No comment. No we have some possibilities; there are interested teams from different countries," Fisher told sports.ru.
"Andrey is on holiday at the moment so we'll discuss the current situation this weekend. Can he go back to Russia? There's a chance for this. But we have to discuss all possibilities and we have to find out what is the best for Andrey.
"He's still very ambitious. He's a great footballer in a good age and ready to play. Money is not the most important criteria. He's ready for new challenges and he's ready to play."
Fisher also insisted that Arshavin was keen on signing for a team that could offer him top level football.
"We have lots of possibilities but I can admit that it is European football that is really interesting," he added.
"Champions League or Europa League - these are great competitions."