Former West Ham defender Bilic, 39, does not want to abandon his country before Euro 2008 despite a chance to manage here in the top flight.
So he is trying to convince Fulham's board he does not have to commit much time to Croatia until the finals begin.
A source close to Fulham admitted: "Slaven has shown all the makings of becoming one of the world's best coaches.
"Obviously he doesn't want to walk out on his country with a major tournament coming up and is trying to convince the club it won't be a problem juggling both roles.
"Fulham know this is a golden opportunity and so may be willing to take that gamble."