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That's according to Match of the Day pundit Danny Murphy, who believes that a Premier League top-four finish wouldn't save the 48-year-old Norwegian.
The Red Devils' hope of ending the season with silverware now lies in the Europa League as Manchester City hold a commanding lead in the Premier League title race.
Solskjaer's men will play Granada in Europe's lower-tier club competition next, but the likes of Arsenal, Ajax and Roma are still in contention for the Europa League title.
Murphy has suggested that United have to be "doing better" and they could "make a change" if they don't win this season's Europa League.
The 44-year-old former Fulham and Tottenham man told talkSPORT: "They can't be blaming tiredness for a loss at Leicester in the FA Cup and their squad not being strong enough.
"United competing in the Europa League is not good enough, first and foremost, the fans know that, the club know that and Ole knows that.
"Not progressing in the Champions League was a massive, massive blow for them. Yes, they're going to get in the top four... but so they damn should!
"He's in a position for me where the Europa League, for me, has become monumental for him and the club.
"I like him and I'm not one who wants to see managers sacked and I do think they've made some strides forward, but Manchester United have to be doing better.
"If they don't win the Europa League and only finish in the top four, then I think they'll make a change."
United will travel to the Nuevo Estadio de Los Carmenes on 8th April for the first-leg quarter-final clash with Granada.
The return leg will be hosted at Old Trafford on 15th April.
Featured Image Credit: PA
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