Rogic has enjoyed a seven-year stay at the Scottish Premiership giants but has often seen injuries reduce his impact on first-team proceedings.
The 27-year-old has been an unused substitute through the Hoops' first two fixtures this season and could soon be the subject of a £4 million (A$7.3 million) move, according to the Daily Mail.
An unnamed Qatari club will hope to get the best out of the Socceroos midfielder, who has made more than 150 appearances for the Bhoys and won a slew of silverware.
Neil Lennon's men are scheduled to face Dundee United on Sunday (AEST), though Rogic's exit could be confirmed before then.
