Sheffield Wednesday boss Danny Rohl is one of the names under consideration for the vacant Southampton role.
There has been no formal approach to the Championship club as things stand.
Two other foreign coaches are also of interest to Southampton, and the Premier League club would not be required to pay compensation for either of them.
Rohl has fond memories of his time working at St Mary's as assistant to Ralph Hasenhuttl. They arrived in December 2018 with Southampton 18th in the league and kept them up.
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Highlights from the Premier League match between Southampton and Tottenham
As a head coach, Rohl also kept Sheffield Wednesday up on the final day of the Championship last season. He formerly worked as a coach at Bayern Munich and the German FA.
Former Brighton U23s and England U19s coach Simon Rusk will take charge of the League Cup tie against Liverpool on Wednesday (_live on Sky Sports Main Event_, kick-off 8pm)
Rusk is also on standby to prepare for the Premier League trip to Fulham on Sunday _(live on Sky Sports Premier League and Main Event,_ kick-off 2pm) if Southampton don't make progress in their search this week.
The first-half performance from Saints against Spurs on Sunday accelerated the decision to sack Russell Martin.
Previously, the club had wanted to give him as much time as possible to turn things around, but it became untenable.