The Blues have reportedly been chasing the player for the last couple of weeks, but now the highly rated midfielder looks poised to join La Liga side Sevilla.
He currently plays for Hungarian side Ferencvaros, who he joined from Hamburg in 2012 and is being courted by a number of clubs in Europe after impressive performances at this summers' World Cup in Brazil.
With a £2m release clause in his contract, Sevilla might look to the midfielder as a replacement for Ivan Rakitic who has joined Barcelona. Swansea City and Sunderland have also been linked with moves for the player, who is comfortable at centre back or in a deep lying midfield position.
Would you like to see Everton sign Muhamed Besic? Vote and comment below.