The 21-year-old, who can play in midfield and defence, impressed at the World Cup, despite Bosnia-Herzegovina's elimination from the tournament at the group stage.
The Ferencvaros midfielder has been monitored by several European clubs, with Sevilla weighing up a move as they target a replacement for Ivan Rakitic, who joined Barcelona earlier this summer.
Swansea City are also interested in Besic, but sources close to the deal have told Goal that Everton are now leading the chase.
It is understood Everton are ready to pay €5 million to land Besic, who left Bundesliga side Hamburg for Ferencvaros in 2012.
Besic started every game as Bosnia-Herzegovina finished third in Group F at the World Cup, picking up three points in their final game against Iran having already been eliminated from the tournament following defeats to Argentina and Nigeria.