The 21-year-old, who represented Bosnia and Herzegovina in the World Cup, has been with the Premier League club on their pre-season tour of Thailand and trained with the squad in Bangkok on Saturday.
"Muhamed is a young footballer who has developed massively in the last two seasons," Everton manager Roberto Martinez told the club's 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 added: “The most important reason why I signed for Everton was Roberto Martinez – I think he is a fantastic coach.
“I know Everton is a really traditional club and the supporters are fantastic here. I think it is the biggest step for me after leaving Hungary.
“I think my style is suited to playing in the Premier League and I think this is the best league in the world.”
Martinez has also set his sights on Anderlecht forward David Henen and is close to adding the 18-year-old to his squad.
The Spaniard said: “David is a young man who is someone with a very good future and we want him to be part of our development squad and get into the Under-21s.
“He’s had a rapid rise in his early years. He’s only 18 and has attracted a lot of attention, but we need to give him a bit of quiet time to allow him to work on his game and develop as a footballer. He has real talent with a view of one day making it into the first team, but that will be down to him and his hard work.”