"Muhamed is a young footballer who has developed massively in the last two seasons," Everton manager Roberto Martinez told the website.
"He has been playing as a centre half and as a defensive midfielder at an incredible level.
"What he did with Bosnia and 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 was born in Berlin and began his professional career with Hamburg, before joining Ferencvaros in 2012.
He played in each of Bosnia's three World Cup games as they went out at the group stage.
(Writing by Stephen Wood, editing by Justin Palmer)
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