Besitkas have edged ahead in the race to sign Yevhen Konoplyanka according to the Turkish newspaper Fanatik.
The report goes onlt state that talks between Konoplyanka and Atletico Madrid, Stoke, Tottenham and Liverpool had all broken down due to the players steep contract demands.
The 25-year-old is understood to be holding out for a €4 million per season deal which includes a minimum fee release clause.
Liverpool have been linked with a move for Konoplyanka in recent weeks by numerous newspapers including the Express.
Beşiktaş finished the Turkish Super Lig season in third place and will play in the Europa League next season.
The Ukrainian international provided seven assists and scored eight goals in 42 appearances for Dnipro in all competitions last season.