The Melbourne City midfielder has been heavily linked with a move to Ibrox, but is said to have alternative offers.
Mooy (L) warms up ahead of Australia's friendly with England
Mooy, who has scored 11 goals and made another 10 for A-League outfit Melbourne City this season, is said to have offers from leading Belgian clubs.
Mooy (L) during a previous spell in Scotland
One thing that could scupper the 25-year-old's Rangers hopes is the Light Blues' recent transfer business, with four midfielders arriving at Ibrox already.
Accrington Stanley pair Josh Windass and Matt Crooks agreed their moves in January, while Jordan Rossiter and Joey Barton have signed for the Scottish Championship winners since their interest in Mooy was first reported.
Barton is one of four midfielder to have signed for Rangers this summer
Sydney-born Mooy, who holds a Dutch passport and therefore wouldn't require a work permit to play in the UK, began his career in England with Bolton Wanderers and previously played in Scotland for St. Mirren.
The 14-cap star impressed during Australia's narrow defeat to the Three Lions on Friday, with his performance arguably overshadowing that of teammate and Celtic midfielder Tom Rogic.
Mooy scores against Harry Redknapp's Jordan in March
He previously admitted he used to wake at 5am to watch Rangers in action.