The 21-year-old midfielder has joined for a fee believed to be worth 5 millon euros, and manager Roberto Martinez is delighted with the capture of the "dynamic" player.
The 21-year-old, who featured for Bosnia-Herzegovina at the World Cup, linked up with his new squad for its summer trip to Thailand last week and has now completed the move that Goal UK understands is worth 5 millon euros.
Besic watched from the sidelines as Everton lost to newly promoted side Leicester City in Bangkok over the weekend and Martinez is delighted to be able to finalize the deal.
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"Muhamed is a young footballer who has developed massively in the last two seasons. He has been playing as a center half and as a defensive midfielder at an incredible level," Martinez told the club's official website. "What he did with Bosnia-Herzegovina in the World Cup, he showed that he is a really mature footballer, that he covers the ground really well, that he is very dynamic and technically very gifted.
"He is bringing us really good strength in an important part of the side."
Besic becomes Everton's second summer signing after fellow defensive midfielder Gareth Barry penned a permanent contract earlier this month.