The 20-year-old was forced off just before half time in the Saints' 1-1 draw at home against West Ham United last weekend after dislocating his shoulder.
There is no timescale on Bella-Kotchap's potential return and Hasenhuttl admits that the centre-back may require surgery as his injury "doesn't look good".
Speaking at a press conference on Tuesday, Hasenhuttl said: "Doesn't look good for Armel, he'll be out on Wednesday. It's not so sure in what direction we are going.
"If we need a surgery for him or if we do it without. I think he wants to try without, mobilising the shoulder, then coming back in maybe two weeks, three weeks. Otherwise it will take longer.
"This is definitely a loss for us, again. So longer term injuries are not the best for a team that is looking for results."
Although Southampton have endured a difficult start to the 2022-23 campaign, winning only two of their first 10 Premier League games and sitting 18th in the table, Bella-Kotchap has made a bright start to life at St Mary's following his summer move from VfL Bochum.
The 6ft 3in centre-back has started nine of Southampton's 10 top-flight fixtures and his impressive performances have been recognised by Germany head coach Hansi Flick, who handed the youngster his first international call-up last month.
However, Bella-Kotchap now appears to be a doubt for the 2022 World Cup in Qatar which begins in the middle of next month.
Bella-Kotchap will now join Valentino Livramento and Romeo Lavia on the treatment table, although the latter is closing in on a recovery from a hamstring injury.
The absence of Bella-Kotchap will likely see fellow summer signing Duje Caleta-Car handed a start at centre-back alongside Mohammed Salisu against Bournemouth.
Meanwhile, Hasenhuttl has also confirmed that Theo Walcott and Ibrahima Diallo are both doubtful for Wednesday's trip to the Vitality Stadium due to illness.
"So far we had only Theo out for the weekend because he was getting ill suddenly on Sunday morning. He should come in today and then we'll see how he is," the Austrian boss added.
"Ibra feels a little unwell today but we will see what is happening in the session today and who is available."
Southampton, who have picked up only one point from a possible 15 available, could climb out of the relegation zone with a win over 10th-placed Bournemouth.