The Cottagers are expected to recruit at least one forward this summer despite the recent impressive form of Rodrigo Muniz.
The Brazilian has scored nine goals in his last 13 appearances to establish himself as Marco Silva's first-choice centre-forward.
However, Fulham still lack depth in the final third, with Chelsea loanee Armando Broja set to return to his parent club this summer.
Carlos Vinicius is likely to follow Broja out the door when he returns from his loan spell with Turkish Super League Galatasaray.
Fulham lining up Kilicsoy move
Raul Jimenez has also struggled to find his best form since arriving from Wolverhampton Wanderers last summer.
The Mexico international has scored just five goals in 22 Premier League matches, highlighting the need for Fulham to bolster their forward options this summer.
According to Sports Digitale, Fulham are plotting a move for Kilicsoy after holding preliminary talks with Besitkas.
The report claims that Fulham's chief scout Brian Talbot has travelled to Istanbul to discuss a potential transfer with Besiktas' technical director Samet Aybaba.
The Cottagers have supposedly set aside £15m for Kilicsoy, but Besiktas have made it clear that the proposal falls below their asking price.
Who is Bestikas' highly-rated youngster?
Kilicsoy is attracting interest from the Cottagers after impressing during his breakout season for the Turkish giants.
The forward, who can also play out wide, has scored 10 goals and provided two assists in 20 Turkish Super Lig matches.
Kilicsoy also made an instant impact for the Turkey Under-21 side, scoring in both of his first two appearances against Georgia and Italy last month.
The young forward is under contract with Besiktas under 2028 after penning a new four-year deal earlier this year.