The 20-year-old has been on trial with the club on their recent trip to Spain and will join the Scottish champions subject to a work permit application.
A host of Premier League clubs, including QPR and Reading, were also said to be keen on the player but the Hoops have snapped up the player with four international caps to his name.
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"The club can confirm an application will be made to the appropriate authorities in the next couple of days."
Rogic has made 24 appearances for A-League side Mariners, scoring five goals.