The Norwegian has overseen a remarkable upturn in fortunes since taking over as caretaker boss in December, replacing Jose Mourinho.
United’s unbeaten Premier League run, which opened with six straight wins, has seen the Red Devils close the gap on the top four, while also reaching the FA Cup quarter-finals.
The change in mood around Old Trafford has seen the calls for former striker Solskjaer to be given the job on a full-time basis.
Austrian Hasenhuttl believes everything appears set up for the transition.
“I think they will (appoint Solskjaer permanently), to be unbeaten to now is amazing and outrageous,” the Southampton manager said.
“With his history as a player (at United) and now doing it as a manager, you can’t have better signs to be the next man.”
Hasenhuttl added: “We spoke about the time of (Sir) Alex Ferguson, where nobody wanted to go there because there was nothing for you.
“In the years after, this myth is not as strong because other teams could take points there. They are going back to the old ways where it is hard to take points from them.”
Southampton boosted their survival hopes with a midweek win over fellow relegation battlers Fulham, which saw them move out of the bottom three.
Hasenhuttl believes his side can now build some upward momentum, despite the significant challenge ahead of getting a result at Old Trafford.
“As a manager and my team, we don’t think this way – we can be a good opponent in every stadium for every team if we are well organised, like against Chelsea, so I am positive,” he added.
“We got a very important win on Wednesday and we can play with not much pressure because no-one expects anything from us.
“I don’t know how they estimate us but we have shown that we can take points from away games.
“It’s fantastic for the players to go there now after Fulham because we can be brave and enjoy the game and maybe make a big surprise.”
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