The Saints succumbed to goals from Joe Willock, Chris Wood, Bruno Guimaraes and Miguel Almiron against the Champions League-chasing Magpies, with Romain Perraud pulling one back.
The 4-1 loss to Newcastle was the latest in a long line of poor results for the Saints this season, and they have only taken 12 points from their opening 14 Premier League matches.
Hasenhuttl's men occupy a spot in the relegation zone ahead of the final Premier League matchday before the World Cup, although only one point separates them from Bournemouth in 17th.
Nevertheless, the Saints have only won one of their nine top-flight fixtures since the start of September, and according to The Athletic, Hasenhuttl's time is up.
The report claims that the 55-year-old is set to be relieved of his duties by the South Coast club, which could happen in the next few days before Southampton face Liverpool next weekend.
Alternatively, Southampton could elect to keep Hasenhuttl in charge until the World Cup break, at which point they will officially dismiss the Austrian.
Southampton had supposedly wanted to give Hasenhuttl more time after the World Cup to turn his side's fortunes around, but recent results have forced a change of heart.
The Saints were booed by their own fans after their defeat to Newcastle, which took Hasenhuttl's record to 60 wins, 38 draws and 75 losses in 173 games in all competitions.
The former RB Leipzig boss arrived in England to replace Mark Hughes in 2018, and in that time he has led Southampton to 16th, 11th and two 15th-placed finishes in the Premier League.
However, Hasenhuttl has also overseen two of the Premier League's joint-heaviest defeats, with Southampton losing 9-0 to both Leicester City and Manchester United under his tutelage.
The Austrian was quizzed on his future by the press after the loss to Newcastle, but he insisted that he was not losing sleep over such speculation, saying: "I'm never concerned.
"I always try to do my job and that's all that interests me. I have taken a lot of decisions since I am here. The good thing is this one I don't have to take."
Should the Austrian be given the boot in the coming weeks, he will become the fifth managerial casualty of the Premier League season after Steven Gerrard, Scott Parker, Thomas Tuchel and Bruno Lage.
It is currently unclear if Southampton already have any replacements in mind, with Hasenhuttl currently preparing his side for their EFL Cup third-round clash with Sheffield Wednesday on Wednesday night.