The 21-year-old represented Bosnia and Herzegovina in the World Cup and has been a target for Everton boss Roberto Martinez.
Although Besic will not feature in a friendly against Leicester in Bangkok, Martinez is pleased to have the former Hamburg man in amongst his current squad.
"The next few days will allow Mo to meet the players," Martinez said on Everton's official website.
"But more importantly it is an opportunity for him to get into optimum condition ahead of the new season.
"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."England international Gareth Barry has already moved to Goodison Park this summer after enjoying a loan spell at Everton last season.