Rogic has one year left on his contract at the end of this current season, and reports from Scotland suggest Celtic are bracing themselves for news that he will not re-sign and instead seek other opportunities elsewhere.
Southampton have been monitoring the 25-year-old for months and consider him a "top summer target" according to BBC Scotland.
Rogic has scored eight goals and made eight assists in 37 appearances this season, including a goal last weekend in a 5-0 victory over Rangers.
The move could prove a stepping stone for the Socceroos midfielder to an even bigger club, like it did for Virgil van Dijk and Victor Wanyama who both moved from Celtic to Southampton before transferring to Liverpool and Tottenham Hotspur respectively.
Southampton are currently in the relegation zone, one point behind Swansea and three points behind both Huddersfield Town and West Ham United.
Celtic manager Brendan Rodgers admitted that the former Central Coast Mariners man could be on his way out of the Scottish champions.
"Tom is a brilliant player, but you have to respect that a player’s career is short," Rodgers said.
“I don’t think he is questioning the club, I don’t think he is questioning anything here.
“He is a young guy who was deemed not good enough for some clubs when he was younger. Tom has come over here and it has taken five years for him to really establish himself and show his real talents.
“In the last couple of years, he has been sensational.
“It’s just a life decision. Tom and his wife are from Australia.
“If you are a Celtic supporter, you probably can’t think of a better place to be in the world than Celtic. But there are warmer and drier places!
“Tom’s a massive talent and we hope we can keep him here. There’s nothing sinister in it.
“He’ll take his time and I’ll always support players, whatever their decisions are.
“Hopefully, he can stay and agree a long-term deal.”
It his understood that Rogic will assess his options after the FIFA World Cup in June - which you can watch LIVE, FREE and in HD on SBS - which could be used as a shop window to showcase his talents to the world.
Rogic has played 128 matches, scoring 30 goals and making 22 assists since joining Celtic in 2013.