The 25-year-old has just one year left on his contract with the Scottish champions and according to 'BBC Scotland', 'Saints' have been monitoring the Australia international for some time.
The Premier League strugglers are believed to have watched Rogic in action during Sunday's 5-0 win over Rangers, a game in which he found the net for the eighth time this season.
A technical player capable of unlocking defences, Rogic has often attracted admiring glances from elsewhere, only for his poor injury record to put sides off.
Southampton have previous for cherry-picking Celtic's best talent, having signed Victory Wanyama from the 'Hoops' in 2013, Fraser Forster in 2014 and Virgil van Dijk in 2015.