Scans on an ankle injury suffered during the 3-1 win over Aston Villa on Sunday revealed ligament damage and Maguire was pictured wearing a protective boot at Old Trafford on Tuesday as United lost 2-1 to Leicester.
The England defender will miss remaining Premier League games against Liverpool, Fulham and Wolves and Solskjaer said it is possible Maguire will not be ready for the Europa League final against Villarreal in Gdansk on May 26.
"The good news is it wasn't broken, there was no fracture, but a ligament injury is serious," Solskjaer said. "If I'm very positive; that is maybe stretching it.
"I'm hopeful he might be ready for it [the final]. I don't think he'll play in the league again but we'll do everything we can to get him ready for the final."
Solskjaer will give Maguire every chance to return before the Europa League final but said he will not take a risk by rushing the 28-year-old back.
"You always try to look after players long-term," Solskjaer added. If he's fit he'll play for us, if he's not he won't. We're hopeful he's ready and if he's ready for us, he'll be ready for the Euros. Norway's not in the Euros, I'm not really that concerned."
Solskjaer faced backlash from some Liverpool fans after picking a much-changed team for the defeat to Leicester which gave Brendan Rodgers' side a boost in the race for the top four.
Bruno Fernandes, Paul Pogba, Marcus Rashford, Luke Shaw and Aaron Wan-Bissaka are expected to return for Liverpool's visit to Old Trafford on Thursday but Solskjaer said he won't be fazed by any criticism from Anfield.
"My job is Man United and my concern is the Man United fans, what they think about my team, what they want from my team and that we come together as one and show what Man United is," he said.