Sheffield Wednesday boss Danny Rohl is one of the names being considered for the vacant Southampton job.
The Championship club has not had any form of formality as things stand.
Southampton are also interested in two other foreign coaches, and the Premier League club would not have to pay compensation for either of them.
Rohl has fond memories of his time working at St Mary's as an 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 head coach, Rohl also held Sheffield Wednesday up on the final day of the Championship last season. He previously worked as a coach at Bayern Munich and the German FA.
Former Brighton U23 and England U19 coach Simon Rusk will take charge of Wednesday's League Cup tie against Liverpool (live on Sky Sports Main Event8pm start)
Rusk is also back to prepare for Sunday's Premier League trip to Fulham (live on Sky Sports Premier League and Main Event, kick-off 2pm) unless Southampton make progress in their search this week.
A first-half performance from Saints against Spurs on Sunday precipitated the decision to sack Russell Martin.
Before that, the club had wanted to give him as much time as possible to turn things around, but it became impossible.
