The 23-year-old midfielder, who has been capped 39 times by Egypt, is thought to have cost around £5million, and will take home £60,000-a-week in wages.
The Egyptian international began his senior career with El Mokawloon in the Egyptian Premier League and has spent three years with Basel, making over 90 league appearances and winning three consecutive Super League titles.
Elneny has been brought in to provide competition and cover for holding midfielder Francis Coquelin.
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The Egyptian midfielder brings title winning experience to an Arsenal side hoping to win the Premier Legaue for the first time in more than a decade.