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Elneny joined Arsenal in January 2016 from Swiss side Basel for around £7.4m but he has been unable to establish himself in Arsene Wenger’s starting XI.
Last season the Egyptian made just 14 Premier League appearances as he struggled to get in the side ahead of Granit Xhaka, Aaron Ramsey and Francis Coquelin and the competition for places mean his chances of more minutes are unlikely to improve.
But Elneny has shown his versality in pre-season so far, covering at centre back with Shkodran Mustafi, Rob Holding and Calum Chambers still on holiday.
According to The Sun, Arsenal were willing to let Elneny leave this summer and the 2016 Premier League champions reached an agreement with them.
But the Egypt international turned down the Foxes after holding talks and is happy to stay despite his situation.
Arsenal are attempting to get a number of players off their wage bill to give Wenger more money to spend on transfers.
Lucas Perez, Kieran Gibbs and Carl Jenkinson are all up for sale.
Mathieu Debuchy had a proposed move to French side Nice collapse earlier this week but Wojciech Szczesny completed his £10m move to Juventus earlier today.