Hasenhuttl is believed to be on the brink at Southampton after another poor start to the season where they have won just two of their opening eight games.
The Saints are 16th in the league table, just a point above the teams in the relegation battle and the Austrian is fighting to save his job.
Steve Cooper was linked with replacing Hasenhuttl but he has signed a new contract with Nottingham Forest.
Marsch was also said to be a contender, but it has been claimed that the Leeds boss is not going to move to the south coast.
The American took charge of Leeds in the middle of last season and helped them survive in the Premier League.
Leeds have made a solid start to the current campaign and the club are in no mood to find a replacement for him.
Marsch may also not be someone Southampton would consider given it would be almost impossible to extract him out of Leeds.
