The 29-year-old full-back, who remains under contract at Ibrox until 2024, has been attracting strong interest from a host of clubs across Europe in recent weeks.
GlasgowWorld understand the reigning Turkish Super Lig champions are currently in the driving seat and are ready to step up their pursuit to land the Croatian international.
Left-back has been a problem position for Abdullah Avci’s side with Polish defender Tymoteusz Puchacz returning to parent club Union Berlin after his loan spell.
Ismail Koybasi has failed to impress in that area and the Turkish side are looking to strengthen as they prepare for a play-off to reach the Champions League group stage.
As of yet, no official offer has been tabled for Barisic, who is also on the radar of an unnamed club in Italy.
The 20-year-old midfielder, who has had several loan spells across the UK in recent seasons, has failed to break into the Light Blues first-team.
The former Scotland Under-19 international made his Gers debut during a Europa League qualifier against St Joseph’s in July 2019 but he hasn’t made a competitive appearance since then.
According to Football Scotland, McPake could move abroad for the first time in his career with Be’er Sheva who finished runners-up in the Israeli Premier League last season.
A temporary switch is in the pipeline, with an option to make the signing permanent likely to be included should he impress.
McPake, who counts Dundee, Morton, Harrogate Town, Morecambe and Tranmere Rovers among his previous loan clubs, has a number of interested suitors, including three Scottish Championship sides.