The Red Devils were dumped out of the Champions League at the group stage on Tuesday night after a 3-2 defeat to Red Bull Leipzig in Germany.
They will now compete in the Europa League and battle to get into the title race in the Premier League, or at least compete for a top four place.
Brown, who won five Premier League and two Champions League titles with United, feels Solskjaer must get his side back into the top four to justify his position, but he will be given until the end of the season to do so.
‘Finishing in the top four is a must for Man United now,’ said Wes Brown, speaking exclusively to Ladbrokes at its launch of the 5-A-Side Bet on Man United v Man City.
‘League position is always the most important thing for me anyway, and nobody would’ve thought United were going to win the Champions League this year.
‘Solskjaer’s now favourite to be sacked first in the Premier League after last night but I can’t see that happening.
United tossed away an excellent position in their Champions League group after beating Paris Saint-Germain and Leipzig in their opening two matches, then needing just a point from their final two, which they failed to achieve.
Premier League form has been better of late, with four straight wins, but they sit sixth in the table behind the likes of Leicester and Southampton.
The Norwegian boss has come under fire as United continue to look off the pace in every competition they compete in, but Brown feels progress is being made and he needs more time to work.
‘He’ll always get stick in the media, but he knows what he’s doing,’ the former England international said. ‘He’s doing his thing, he’s getting the players playing the way he wants them to. He’s trying to build a really strong squad, because you need exactly that in this league with so many competitions you’re playing in.
‘He gets a lot of criticism, but that’s Manchester United for you. If you do get beat, you’re going to get stick, and if you win, you’re going to get praise. You could probably say there’s a lot more experienced managers who’ve had the job since Sir Alex Ferguson who got as much criticism and things didn’t work out for them.
‘I don’t think Ole is listening to any of the criticism, to be honest. I think he’s more concerned about getting the team back to winning ways on a more consistent basis and getting the squad as strong as he can.
‘Ole understands the club. He’s been a manager for a while now and he knows where he wants to put the club, back where it belongs. We all understand that doesn’t just happen overnight.’
Former Tottenham boss Mauricio Pochettino is the man most heavily linked with replacing Solskjaer and, although he is not pushing for it now, Brown admits that he wouldn’t be surprised to see him at Old Trafford one day.
‘Is Mauricio Pochettino holding out for the Man United job?’ Brown queried. ‘You’d have to ask him. I think Ole’s doing a good enough job at the moment. You can never say never to Pochettino at United because that’s football, but Ole won’t be thinking about that at all.’