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.
Martinez is delighted to have Besic on the trip and is now keen to reach an agreement with Hungarian side Ferencvaros for his transfer to Goodison Park.
"We hope things will be wrapped up for Besic soon. We have a good relationship with Ferencvaros, which has been really open and strong throughout,” Martinez told his club’s official website.
“There are certain details that need to be solved and once that’s done we’ll be okay.
"Mo already feels like part of the squad. It’s been so important for him to travel with us, interact with the squad and get to know us all.
“He trained on Saturday and that was a real positive. He’s also getting the right balance of recovering from the World Cup. He’s only 21 but it’s incredible how these bodies recover.”
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.”