The Norwegian is still yet to win any trophy of note since his appointment in 2018 and under immense pressure to end the club's silverware drought, having seen another chance slip through his fingers on Wednesday night.
Solskjaer has faced questions over whether he's the right man to lead the Red Devils throughout his tenure, and those only continued after being dumped out of the cup by the Hammers at Old Trafford.
If the results wasn't enough, fans weren't happy after the boss gave a less than encouraging post-match interview, saying the Red Devils 'gave it a go' against David Moyes' team, which has only amplified the scrutiny.
The Norwegian made 11 changes to his Man United team, but that was hardly an excuse for the result, as Moyes made ten.
"It was a poor start again, that needs addressing," Solskjaer told Sky Sports.
"After that we played well for long periods and gave it a good go. Stats don't mean anything but when you've got 27 shots you know the boys have given it a good go."
Outspoken talkSPORT pundit and former Crystal Palace owner Jordan chose not to hammer Solskjaer after Manchester United's defeat to Young Boys in their Champions League opener earlier this month.
But he didn't pass up the chance a second time, as he claimed United are now no longer an 'elite side' with the former striker at the wheel.
The White and Jordan host said: "Are Man United an elite side? They're an elite football club but elite sides win something, they haven't won anything for nearly half a decade.
"Does that make them an elite side or an elite football club? Economically they're an elite football club, they don't have an elite manager. I know people don't like to hear that, but they don't have an elite manager.
"An elite manager doesn't look like Ole Gunnar Solskjaer, he looks like Thomas Tuchel or Pep Guardiola and his achievements look like something very different, i.e. winning something in the last three or four years.
"They don't have elite performances right now. Winning a League Cup to go to your central theme, well if it's good enough for Manchester City and good enough for arguably the best team over the last decade in English football to make that a stable diet, then it would defeat the naysayers and people like me that Ole Gunnar Solskjaer doesn't win anything.
"Winning something is better than winning nothing, as it is we are 12 tournaments down of which his record is for 12 currently.
"Three years, each time he gets an opportunity to play in four tournaments, he's had 12 tournaments and he's won none of them.
"Is that what Manchester United signed up for as fans, is that the direction of travel? Is it good enough to build a side that's competitive?
"If you spend this amount of money how can it not be a competitive side?
It's a difference between competitive and contenders and winning, and being a contender - which he is - for making a competitive side that looks again as if it's going on a journey of winning very little."
Alongside Jordan's comments, fans phoned into talkSPORT's Sports Bar show on Wednesday night to vent their frustrations.
"Ole Gunnar Solskjaer is reliant on Mike Phelan and the back room staff and it's just not good enough," said one peeved United supporter.
"He's playing [Jadon] Sancho out of position, he played out on the right last year, we've brought him in, he's played on the left and it's just not good enough.
"Ole's not tactically astute, he's not well versed in that, he's come from Cardiff, he's come from Molde and he's then come into the club.
"With Solskjaer, there's no plan B, we've seen this time and time again, he hasn't got a plan B, he can't rely on anyone to bring on and we suffer and that's why we haven't got the trophies.
"Ole's a legend, but I'm starting to worry about his decisions. I can't figure it out, League Cup, that's a back-up for any manager for me.
"[Man] City have won it four years in a row, I feel like that's what we used to do, Fergie used to make sure we get that League Cup and then we could fall back on it."