The Saints have made a disappointing start to the Premier League campaign, taking just nine points from 14 games.
That tally leaves them inside the relegation zone and the south-coast club have taken the decision to bring in a new boss in an attempt to stave off another battle against the drop.
The 51-year-old left RB Leipzig in the summer and reports suggest that the Austrian is Southampton's first choice in terms of a permanent replacement for Hughes.
“It has pretty much been one way traffic for Ralph Hassenhutl and his price has tumbled considerably in less than an hour,” said William Hill spokesman Joe Crilly.
In two years at Leipzig Hasenhutti had a 48 per cent win record and prior to that role he spent time at Ingolstadt, Aalen and Unterhaching. His high tempo approach would likely be a welcome change compared to that which has seen Southampton score just 12 times thus far.
Paolo Sousa was pictured in the stands when Southampton took on Fulham recently, prompting rumours that the former Swansea and Leicester boss would also be a contender should the job become available. The Portuguese is 6/1 (7.0) second favourite at the time of writing.
The former Everton and West Ham boss is often heavily linked with jobs where avoiding relegation is the primary objective but the prices suggest Southampton are looking away from the usual candidates such as Allardyce, David Moyes (33/1) and Alan Pardew (25/1).
With Southampton facing a crunch clash against Cardiff City at the weekend, a swift apppointment may be on the cards and Hasenhutti is certainly the frontrunner according to the odds compilers.