According to his agent, Elneny has been away with his country and has contracted the disease, entering a period of self-isolation with team mate Salah.
His agent Alan Nazmy posted on Twitter, "Elneny Positive Cov19 get well soon."
The 28-year-old played against Togo in an Africa Cup of Nations qualifier on Saturday.
Elneny is likely to miss Arsenal's trip to Leeds on Sunday, as well as their Europa League clash at Molde next week.
Elneny has enjoyed a run in the team at Arsenal this season having spent the last campaign on loan at Besiktas.
He has played 11 times for the Gunners in all competitions this term under Mikel Arteta.
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