The 26-year-old, who just recently lifted his 10th trophy with the Scottish giants, is being targeted by the two Premier League sides, reports The Sydney Morning Herald.
Southampton were linked to Rogic in May 2018 ahead of the FIFA World Cup, but no deal ever eventuated.
The Saints escaped relegation this season after German coach Ralph Hasenhüttl took over in December with the club not having won a match in three months.
Should he join the club, Rogic would be competing with central midfielders Stuart Armstrong, Oriol Romeu, Pierre-Emile Hojbjerg and James Ward-Prowse for a position in the starting XI.
Rogic played 33 matches this season for Celtic, scoring five goals and producing six assists.
The Socceroos midfielder still has four-years left on his deal with the Scottish club, meaning a bid would have to be significant for the deal to go ahead.
The report also linked Rogic with his former coach at Celtic, Brendan Rodgers, who is now at Leicester.
The Foxes are currently in ninth position on the table with one match remaining.
Should Rogic link up with Rodgers, he would be up against Wilfred Ndidi, James Maddison, Demarai Gray and potentially Youri Tielemans who has impressed while on loan from Monaco.
Rogic has always maintained publicly that he is happy at Celtic, however a move to the Premier League could prove tempting.
