The Egyptian midfielder spent the 2019/20 season on loan with the Turkish side and could have rejoined had the two parties agreed a deal.
Elneny featured 36 times for Besiktas, before playing 41 times during the 2020/21 campaign as he returned to Arsenal.
Fellow Turkish side Galatasaray were also interested in the midfielder's services but the move broke down for the same reasons.
The club's president Burak Elmas vetoed the move himself after Elneny demanded a salary reportedly worth €2.5million (£2.15m) a year over a four-year-contract.
It would have meant that Galatasaray would have had to pay Elneny €15m (£12.87m) over the course of the deal which they were unwilling to do.
The Egyptian midfielder only has one year left on his contract and unless someone else comes in for him in the next few days, Arsenal will try to move him on again in the January transfer window.
The worst case scenario for the club is that he leaves as a free agent next summer and with Granit Xhaka, Thomas Partey, Albert Sambi Lokonga all ahead of him in the pecking order it seems unlikely that he will feature much this season.
Elneny has been at the club since January 2016 and featured 132 times in all competitions for the club and has helped them win one FA Cup and two Community Shields.
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