Saints decided to axe Argentine Pellegrino on Monday, with the club just one point above the relegation places.
Rodgers has impressed since moving to Celtic and reports in Scotland say Southampton are interested in offering the former Liverpool boss the chance to return to the Premier League.
It is not the first time the Northern Irishman has been linked with a top-flight English club since signing on at Parkhead however, and previously Rodgers has played down any suggestion he could be open to another Premier League move.
Mark Hughes and Marco Silva have also been linked with Southampton and the south coast club could focus on one of those two managers rather than Rodgers, as both Hughes and Silva are currently unattached.
Burton midfielder Martin Samuelsen will use the international break to recover from illness after pulling out of Norway U21's games.
Jonas Lossl is staying 'optimistic' over Huddersfield's chances of survival this season and has urged his team-mates to 'go for it'.