The report states that Besiktas have agreed a €2.5 million per season deal with the player and are close to agreeing a fee with the London based club.
The Black Eagles are understood to have set aside €4 million for the Greek international who is expected to leave London this summer. Fulham are keen to sell the Greek international who is on high wages at the London based club.
Mirtoglou has scored 19 goals in 32 appearances this season while on loan at Olympiakos. However, the Greek side do not have the recourses to buy the 27-year-old on a permanent basis.
The Black Eagles are on the hunt for a new striker after selling Demba Ba to the Chinese club Shanghai Shenhua for €13 million.
Beşiktaş finished the Turkish Super Lig season in third place and will play in the Europa League next season.