Tilio, a product of Sydney FC’s youth academy, moved to Melbourne City in 2020 and has since become an important member of their first-team squad.
He has managed 82 appearances for the Melbourne side so far this season, making 19 goal contributions and is attracting interest from Scotland.
Former manager Postecoglou, an Australian himself, first floated the name as a possibility, but he has now departed for Tottenham Hotspur.
Now Rodgers has asked Celtic if a deal for Tilio can be pushed through.
Rodgers is assessing the Celtic squad he has inherited and it appears he believes that Australia international Tilio would be a good fit.
The former Leicester City boss is keen on adding more depth to his attacking line-up, which consists of Kyogo Furuhashi and Oh Hyeon-gyu.
Tilio earned his international debut for Australia just last year and a switch to Celtic would be a big moment in the 21-year-old’s progression.