Barkas has only played once for Celtic in the past five months since the arrival of Joe Hart and appears behind fellow reserve stopper Scott Bain in the pecking order.
The 27-year-old joined Celtic from AEK in his homeland in the summer of 2020 for £5million, but his performances last season were underwhelming.
Greek and Turkish clubs have been linked with a move for Barkas, who is under contract until 2024, in the January window and Postecoglou says that his future may be “elsewhere”.
"Vasilis is a hard worker,” Postecoglou said in an interview with a Greek TV station. “He tried to settle in, but it was difficult with Covid situation.
"But he is a hard worker. We will see what will happens at Celtic or elsewhere."