Celtic made a firm offer early Wednesday morning (AEDT) which was accepted by the cash-strapped A-League club.
The transfer fee is understood to be in the region of $700,000.
Rogic was on a trial with the Hoops at a winter camp in Malaga for a week while he was serving a suspension for getting himself sent off in the match against Sydney FC on 27 December and he left a favourable impression on manager Neil Lennon.
The attacking midfielder will travel to Glasgow from Spain with the team and discuss personal terms with the club over the coming days.
He is likely to be offered a three-year deal worth at least $15,000 a week.
Subject to Rogic reaching a personal agreement with Celtic, the attacking midfielder will formally be transferred to Lennon's team and will be available to face Juventus in the round of the 16 of the UEFA Champions League.
Celtic must register its Champions League squad by 1 February.
"This is hopefully a big step for the player and a good deal for the Mariners so it's a win-win for everyone," Mariners general manager of football Lawrie McKinna said.
"Celtic is a great club with a massive tradition so this is a great opportunity for Tomas.
"We are happy with the deal because he would have been off contract in three months but it is also important that Tom himself is very happy too, which I think he is."
Coach Graham Arnold said he was disappointed to lose such a quality player but felt proud for him and the club.
"He's a lovely bloke and of course I'm sorry to see him go and that he got a red card in his last match in Australia," Arnold said.
"But I'm proud for the club that has produced guys like Mustafa Amini who has ended up at Borussia Dortmund and Rogic who is set to join Celtic."
Rogic was refused permission to play for Celtic in a trial match against Romania's Steaua Bucharest in Malaga because of insurance problems. Celtic won 2-1.
It's another big step in the dramatic rise of 20-year-old Rogic who has only held a professional contract for one year and made his Australia debut last month.