The Saints sacked Nathan Jones over the weekend after a dispiriting loss to Wolves at home on Saturday.
Southampton are in the market for a new manager and the club are pushing to put a new face in charge of the club as soon as possible.
Marsch, who was sacked by Leeds earlier this month, has been on the radar of Southampton for a long time.
And it has been claimed that the American is very much the frontrunner to become the next manager at St. Mary’s.
The Saints have been looking at Marsch for a while and are impressed with the way he kept Leeds in the Premier League last season.
The American held talks with the club this week and Southampton are pushing to sort out a deal soon.
Saints want a new manager in place before the team visit Stamford Bridge to take on Chelsea next Saturday.
It remains to be seen whether Marsch would be keen to take charge of a side who are rooted at the bottom of the Premier League.
