Midfielder Besic, 21, rose to prominence at the FIFA World Cup in Brazil, as he represented Bosnia-Herzegovina in their maiden finals appearance.
Besic moves to Everton from Hungarian side Ferencvaros, and the 12-time Bosnia international said his decision to join the Merseyside outfit was largely thanks to Spanish boss Martinez.
"The most important reason why I signed for Everton was Roberto Martinez – I think he is a fantastic coach," Besic told evertontv.
"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."
Besic's first assignment at the World Cup was keeping a check on three-time Ballon d'Or winner and eventual Golden Ball holder Lionel Messi.
And the German-born Besic said he felt his performance in the fixture against eventual finalists Argentina in Rio de Janeiro - a fixture Bosnia lost 2-1 - thrust his name into the headlights.
"I think my style is suited to playing in the Premier League and I think this is the best league in the world," he said.
"It will be difficult for me because we have fantastic players but I think, with the help of the coach, I will do my best.
"The World Cup was really important and I think the most important game for me was the Argentina game. I think I did a good job and many clubs were looking, including Mr Martinez.
"I can play anywhere [in defence or midfield] but I think I will play as a central midfielder for Everton.
"I enjoy playing in midfield because I get to see a lot of the ball and I think it is my best position."
Besic joined Everton on a five-year deal, for a reported transfer fee of £4 million.
