The Norwegian's future in the dug-out is under the spotlight once again after his side suffered their heaviest ever defeat to their arch-rivals at Old Trafford.
Solskjaer came out swinging in his post-match interview, claiming he will not resign from his post despite the growing calls to do so.
He said: "I have come too far, we have come too far as a group. We are too close to give up now.
"It is going to be a difficult one. The players will be low but there's loads of characters there.
"We know we are rock bottom, we can't feel any worse than this. Let's see where we take it."
However, the choice may be out of his hands whether he can continue with Zinedine Zidane currently the favourite to succeed Solskjaer in the wake of the result.
Could Solskjaer be sacked?
The former Cardiff manager's job was given reassurances about his position after last weekend's 4-2 defeat to Leicester.
According to The Athletic, Solskjaer has 'a lot of credit in the bank at board level'.
The Sun also report that the United hierarchy are continuing to back their manager.
It was said that some agents have offered their services to the Red Devils in a search of a replacement but the club rebuffed their advances.
However, it remains to be seen whether that stance has changed after the 5-0 defeat to Liverpool on Sunday.
What has been said?
Micky Gray said on talkSPORT: "It's back to the drawing board for Manchester United after this. I don't know where they go, what direction they take but the Man United fans won't take this every week.
"They're a loyal bunch, aren't they? They stick to their own, Solskjaer has said they are Manchester United people but if it's not bringing you success then it might not have to be a Manchester United person.
"I just think it's come to the end now.
"You can have all the ability in the world… but the players stopped today. There was nothing left after 50 minutes, Liverpool took the fight out of them and were absolutely first class.
"I've been to this stadium a lot over the last decade. I don't think it's been as bad as that, as empty as that."
Out of principle, Gary Neville never calls for a manager to get the chop but his comments about his former teammate after the drubbing were very telling.
He said: "This is a monstrous day for Manchester United losing 5-0 at home to Liverpool.
"The board are unequivocal in their support of the manager but that's going to put an enormous amount of pressure on the board.
"The fans are not going to accept that… they're always outstanding but Manchester United fans know when it's not acceptable and that's not acceptable.
"Something has got to change in that dressing room and with that coach very quick. They can't keep losing games like that."
Neville did offer a glimmer of hope for Solskjaer as he added: "If this was in the post Sir Alex [Ferguson] aftermath, Ole would be in massive trouble tonight.
"I think the club will hold their nerve, I don't think they've planned for another manager and they'll sit with Ole until the end of the season."
Fellow pundit Jamie Carragher took a different view, as he said: "I think Ole has done a good job but to take on Klopp, Tuchel and Pep, Manchester United need a better manager.
"I wasn't saying that on Monday in a disrespectful way, I feel for Ole - that will be the darkest day in his career.
"I don't want to get involved in a slanging match, I know Ole said something back and rightly so.
"I'm not comfortable saying someone should be sacked - but we are comfortable saying Fred is not good enough or Scott McTominay is not good enough so I do not feel too bad saying Ole Gunnar Solskjaer is not good enough.
"Manchester United need a better manager. Ole is not a manager for Manchester United.
"He's done really well but you need someone to take them to the next level. Unfortunately, Solskjaer is not Jurgen Klopp, Pep Guardiola or Thomas Tuchel. It's staring you in the face."
Who could replace Solskjaer?
Former Real Madrid boss Zinedine Zidane, Leicester's Brendan Rodgers and ex-Chelsea chief Antiono Conte are among the frontrunners to replace Solskjaer should he be shown the door.
The likes of Mauricio Pochettino, Erik ten Hag and Laurent Blanc are also in the mix according to the bookies.
Even England manager Gareth Southgate has been mooted as a potential outside pick for the hot-seat at Old Trafford.