Solskjaer has endured a mixed spell in the Old Trafford hot-seat since he replaced Jose Mourinho midway through the 2018-19 season.
The Red Devils finished third in the Premier League last term and reached the Europa League semi-finals, but have struggled to maintain consistency during Solskjaer’s reign.
Manchester United suffered a shock opening-day defeat to Crystal Palace last week, while they were fortunate to beat Brighton on Saturday.
Keane admits he has been underwhelmed by United’s start to the season and says Solskjaer will likely be axed if the team fall out of the Champions League places.
Asked on Sky Sports’ Monday Night Football whether he believes Manchester United would sack Solskjaer if he fails to secure a top-four spot, Keane said: ‘Yeah, I think so.
‘You’re Manchester United manager. If they can invest in the next week with one or two signings… but if they don’t, it’s tough days ahead for United.
‘The remit will be different for Ole this year. Last year it was all about getting top-four.
‘It will be higher this year. They need to get closer, the points total can’t be the same.
‘Ole needs to get his hands on a trophy because the honeymoon period is over. The pressure is on now.’
Manchester United suffered a 3-1 defeat at home to Crystal Palace last week but returned to winning ways on Saturday with a dramatic 3-2 win at Brighton.
Despite the victory, Keane says the ‘signs are not great’ for Manchester United and admits he is worried about how many goals and chances the team are conceding.
Asked to rate United’s start to the season, Keane added: ‘Not great. They would have pleased with the result against Brighton, but there’s worrying signs already.
‘It’s early days but they’re giving up chances and goals. The signs aren’t great. The amount of chances they’ve given up is a worry.
‘You would expect them to do better against Palace and Brighton. As a whole, it’s not been great.’