Saints vice-chairman of football, Les Reed, is a big admirer of Rodgers and is hoping to convince owner Gao Jisheng to make a move for the Northern Irishman.
Whether the Northern Irishman would be tempted into leaving the current Scottish Premiership leaders for a team just one-point outside the Premier League relegation zone following a 3-0 defeat to Newcastle United remains to be seen.
Who could come with him?
If Rodgers does succeed Pellegrino at St Mary's would the 45-year-old then look to bring some of his current Bhoys players with him to England?
Well someone that the Hampshire club have been linked with before is Moussa Dembele, with the Mirror reporting in late December that they were considering a £20million move for the striker during the January transfer window.
The 21-year-old has scored 45 goals in 80 appearances in all competitions for the Hoops since a 2016 switch from Fulham and perhaps Rodgers could be the man to finally bring Dembele to the top flight of English football.