Marsch was relieved of his duties by Leeds last Monday after a string of poor results had left them facing a relegation battle along with Southampton.
Jones, meanwhile, parted ways with the south coast club on Sunday after just 14 games and three months in charge.
Saints are currently rock-bottom of the Premier League with just 15 points and four wins all season.
They will be hoping Marsch can salvage their campaign with a rescue mission reminiscent of the job he performed at Leeds last term when he guided them to league safety on the last day of the season.