The Egypt international enjoyed an impressive loan with Besiktas last season, leaving a positive impression.
And with his contract now inside it's final 18 months, Besiktas are exploring the prospect of buying Elneny this month.
And the push has the backing of Besiktas coach Sergen Yalcin, who has stated he'd have kept Elneny for this campaign if given the chance.
This season, Elneny has impressed under manager Mikel Arteta, finding a place in the Basque's first team plans. So far this term, Elneny has managed 19 appearances - including ten in the Premier League.
The Egyptian also famously scored this season with a screamer in Arsenal's Europa League win at Dundalk of the Republic of Ireland.
With Besiktas last term, Elneny managed an impressive 36 appearances, scoring once and providing four assists.
- updated January 4
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