Ralph Hasenhuttl takes the Saints to Old Trafford on Saturday afternoon to face a team in scintillating form.
Since Solskjaer took over from the sacked Jose Mourinho last December, he has led them to nine wins and two draws in 11 Premier League games.
The Norwegian initially took over as interim boss and was expected to return to Molde at the end of the season.
However, he is now the frontrunner to get the job full-time and Hasenhuttl believes he must be kept on.
“I think they will [give him the job],” the Austrian said.
“To be unbeaten [in the Premier League] since he took the job is amazing, it’s outrageous.
“And I think with his history of being a player at Man Utd and to do it now as a manager in a very good way, I think you can’t have better signs to be the next manager.”
Southampton are looking for back-to-back wins after a 2-0 victory at home to Fulham on Wednesday lifted them out of the relegation zone.
Solskjaer says they are well prepared going into the game and is hoping for a win in front of their supporters after they were held to a 0-0 draw by Liverpool last time out.
“We’ve had a good look at them,” he said.
“You’ve gotta be two or three games ahead, it was a big win against Fulham, he [Hasenhuttl] did great when he came in, they had a tougher spell in February but when you win a game like this you’re confident.
“We need to get back to winning ways at home and Southampton might change but we’ve got to focus on what we improve on.”
Attention turns to their Champions League last-16 clash with Paris Saint-Germain immediately after the Saints clash, with United 2-0 down from the first leg.
However, Solskjaer insists they will not have one eye on that tie.
“Wednesday doesn’t come in mind whatsoever,” he added.
“Saturday, 3 o’clock, Wednesday night in Paris, we’ve got so much time to get our recovery in, it’s more about Saturday afternoon and see if everybody is available again, 100% focused on this one.”