The 20-year-old will officially become a Celtic player if he is granted a work permit.
A Celtic statement read: "The club can confirm an application will be made to the appropriate authorities in the next couple of days."
Rogic has scored five goals in 24 A-League appearances for Central Coast Mariners and impressed Celtic manager Neil Lennon in a trial spell during their recent Spanish training camp.
The Australia international flew into Scotland yesterday with the rest of the Celtic squad after the Mariners accepted a bid believed to be around £400,000.
The player's future has been the leading A-League transfer story, with Barclays Premier League clubs Fulham, QPR and Reading having also taken an interest.
Rogic will join Rami Gershon in signing for Celtic this month, with the Israeli defender having arrived on loan from Standard Liege.