The 26-year-old has scored 32 goals this season in what has become a prolific campaign in the Turkish Super Lig.
Yilmaz's representatives were reportedly close to agreeing a deal with the Premier League side last month.
However, it fell through at the last-minute due to an inflated signing-on fee.
Fulham are now confident that a switch can be put in place for the player, according to The Metro.
Yilmaz is expected to move to Craven Cottage for around £5m.
Meanwhile, forward Andy Johnson has been linked with a move away from the Premier League club.