The South Coast club are on the lookout for a new boss after sacking Mauricio Pellegrino on Monday evening.
The Argentine leaves with the Saints just one point above the relegation zone after failure to win their last four league games.
According to the Scottish Sun, vice-chairman Les Reed is said to be an admirer of the Celtic boss and will push for his consideration.
Mark Hughes is the current odds-on bookmakers’ favourite, while former Watford and Hull City boss Marco Silva is also said to be in the hunt.
Rodgers is currently on a four-year, £2.5million-per-season contract at Parkhead and has previously stated his long-term commitment to the Ladbrokes Premiership leaders.