The Celtic boss is one of several high-profile names to have been linked with the post as pressure mounts on Mark Hughes, who has gone seven games without a win.
The Saints sit 16th in the table, just two points ahead of Fulham and four ahead of Huddersfield, who play each other tonight.
Manchester City helped themselves to a 6-1 win at the weekend, with Danny Ings scoring Saints’ first goal in six matches.
Hughes has seen his side win just one league fixture this season; a 2-0 away win at Crystal Palace at the start of September.
The St Mary’s side have struggled in recent weeks, suffering a 3-0 defeat at home to Chelsea, a 2-0 loss away to Wolves and a 3-0 reversal against Liverpool at Anfield.
Former Everton, West Ham and Manchester United boss David Moyes is the current favourite, with another ex-Toffees boss, Sam Allardyce, the second favourite.
Republic of Ireland boss and ex-Celtic manager Martin O’Neill is 7/1, along with Rodgers, while Belgium boss Roberto Martinez is priced at 10/1 along with his Northern Ireland counterpart Michael O’Neill and ex-Palace manager Alan Pardew.
Louis van Gaal and former Scotland boss Gordon Strachan, who led Southampton to the FA Cup final and Europe in the early 2000s, are priced at 12/1 and 16/1 respectively.