The Southampton boss has earned praise for engineering their recovery from a 9-0 defeat to Leicester last season to top of the Premier League table, while pressure is mounting on Solskjaer after another poor start to the campaign from United.
And talkSPORT has been told Hasenhuttl can 'follow in Mauricio Pochettino's footsteps' and leave St Mary's to manage one of the top clubs in England.
The Saints have come a long way since suffering that mammoth defeat to the Foxes last season.
That humiliating loss at home left Hasenhuttl on the brink of the sack, but the club stuck with him and he was later hailed for guiding the south coast outfit to an 11th place finish in the Premier League.
They've gone from strength to strength at St Mary's in the 12 months since, and beat Newcastle on Friday night to sit top of the league - albeit briefly ahead of the weekend's action.
STOP THE COUNT pic.twitter.com/rS94knWEhO
— Southampton FC (@SouthamptonFC) November 6, 2020
United, meanwhile, have not enjoyed a good start to the season and if they lose to Everton - live on talkSPORT - could be dangerously close to the relegation zone if results don't go in their favour on Sunday.
It follows their dreadful start to last season, too, which was officially the worst in the club's history, and it means Solskjaer is again facing major doubts over his future in charge.
And speaking in October, talkSPORT reporter Alex Crook believes Hasenhuttl's calming influence and positive attitude is just what United need to recover from nearly eight years of disappointment since Sir Alex Ferguson's retirement from the hotseat.
It's a job Pochettino has also been linked with.
"With my south coast hat on you've got to praise Ralph Hasenhuttl - King Ralph as I've christened him - because I thought it was a tactical masterclass against Everton," said our south coast correspondent.
"The one positive from the behind-closed-doors experience is you get to hear exactly what's being said on the pitch and from the coaching staff, and he cajoled that team, he encouraged his Southampton side from the first minute to the last.
"It was a coaching masterclass, they were well-drilled, they knew how to negate Everton's strengths, he had Danny Ings dropping into midfield much like Harry Kane has done for Tottenham in recent weeks.
"I would take Ralph at Manchester United tomorrow. I think he's a genius.
"You know why? On this day a year ago, exactly a year to the day, Southampton lost 9-0 at home to Leicester. There aren't many managers who would have bounced back from that.
"And not only has he bounced back, he's gone from a position of strength now, they've got the eighth best record in the Premier League since that 9-0, and it's the same group of players.
"He's a good, old-fashioned, decent training ground coach.
"He 's got the personality and the aura about him to handle the job at Manchester United, absolutely.
"He's a big man in terms of stature as well, he's huge actually [he's 6ft 2in].
"I don't think he'd be intimidated by managing those huge names, I really don't.
"I think could well follow in Pochettino footsteps and go and manage one of the big clubs in the country."
Ralph Hasenhuttl has been described a 'genius' manager who would be a perfect fit to replace Ole Gunnar Solskjaer as Manchester United's next manager.