Besic was one of the bright spots of a disappointing maiden World Cup campaign for Bosnia, which ended in a group-stage exit.
The 21-year-old has been heavily linked with a move the Goodison Park club, whose only signing of the transfer window has been a permanent deal for midfielder Gareth Barry.
Martinez confirmed earlier this week that Everton were close to a deal for highly rated Anderlecht forward David Henen.
And the Spaniard has now all but confirmed the capture of Besic from Hungarian side Ferencvaros, and hopes to present him as an Everton player when the team returns from a pre-season trip to Thailand next week.
"The next few days will allow Mo to meet the players, but more importantly it is an opportunity for him to get into optimum condition ahead of the new season," Martinez told the club's official website.
"I am looking forward to introducing him to the Everton fans when we come back but first I would like to thank Ferencvaros for the way in which they have approached the negotiations.
"They have been very understanding at this point and we have enjoyed a good working relationship."