Besiktas have already given up on Liverpool goalkeeper Loris Karius.
The German is on a two-season loan with the Turks, but has failed to win over management.
Fotospur says Besiktas will re-sign goalkeeper Fabri from Fulham in the summer to replace Karius.
Fabri joined the Cottagers from Besiktas last year but will seek a move if his current club are relegated to the Championship.
The Turkish outfit see their former stopper as an upgrade to the Liverpool stopper.