Solskjaer has endured a turbulent spell in the Old Trafford hot-seat since replacing former manager Jose Mourinho in December 2018.
Under Solskjaer, the Red Devils enjoyed a superb second-half to the 2019-20 season, climbing to third in the Premier League and reaching the Europa League semi-finals.
Manchester United have made a poor start to the new campaign, however, losing at home to Crystal Palace and suffering a 6-1 defeat to Tottenham before the international break.
Murphy cannot see Man United securing a top-four finish in the Premier League this season and believes Solskjaer will be sacked before the campaign finishes.
He told the Football Index Podcast: ‘Manchester United have had a very shaky start to the season and I don’t see them qualifying for the Champions League this season.
‘I don’t expect Solskjaer to survive the season as Manchester United manager. I like Solskjaer, but the recruitment has let him down and I expect him to pay the price.
‘I think the balance of the squad isn’t right defensively and they’re far too easy to play against.
‘Their captain Harry Maguire has looked well short of his best and obviously has distractions off the pitch. Even Wan-Bissaka looks like a shadow of the player we saw making tackles for the club last season.
‘They simply won’t be competitive while they concede goals as they do.’
Former Premier League and England star Murphy added: ‘Manchester United had an incredible second half of the season last year and they did very well to secure third position with a place in the Champions League.
‘I wonder if a few of the players started to believe their own hype and that has led to them underperforming this season.
‘I think they need to go back to putting in the hard yards and showing the desire to outwork their opponents again.’
Manchester United, who are currently only 16th in the Premier League, travel to St James’ Park on Saturday to face Newcastle United.