Arsenal signed defensive midfielder Elneny from Basel for £5m in January 2016.
He has since made 38 appearances for the Gunners, many of which have come as a substitute or in cup competitions.
The Sun claim Leicester are keen to sign the 25-year-old this summer and Arsenal have agreed to a £10m sale.
But it is said that Elneny has rejected the chance to move to the King Power Stadium.
The Sun say that Egyptian held talks with the Foxes but turned them down as he loves life in London.
Despite his limited Premier League playing time, Elneny is reportedly happy to remain in his bit-part role at Arsenal.
Arsene Wenger was willing to let him go due to Arsenal’s wealth of midfield options.
And The Sun add that Leicester could make a second attempt at convincing Elneny into a summer switch ahead of the new season.
Elneny played a full 90 minutes in Arsenal’s pre-season clash against Bayern Munich earlier today.
The Gunners earned some revenge over the German giants after claiming a penalty shootout win in Shanghai.
