Eredivisie side Utrecht are said to be preparing an offer for the 28-year-old stopper, who has struggled at Celtic since his £4.5million move from AEK Athens two years ago.
Barkas has made 24 appearances for the Hoops but played just twice under Ange Postecoglou last season after falling completely out of the first team picture following the arrival of Joe Hart.
The club are keen to offload the Greek international this summer with the Daily Mail claiming that Dutch side Utrecht could be ready to make a move for the player who still has two years left on his Hoops contract.
Celtic have been linked with a move for Bosnian goalkeeper Nikola Vasilj but any deal is likely to hinge on Barkas' departure with the club currently well stocked in that department.
Hart and back-up Scott Bain both under contract until 2024, while former number one Conor Hazard - currently on loan with HJK Helsinki in Finland - and highly-rated youngster Tobi Oluwayemi are also on the books.
Ross Doohan, who spent last season on loan at Tranmere, is out of contract this summer and could be set to move on, while another young goalkeeper, Ryan Mullen, has been released with Sheffield United credited with an interest.