Maguire landed awkwardly during the win over Aston Villa on Sunday and was subsequently found to have ankle ligament damage.
United are due to face Villarreal in the Europa League final on May 26, while England’s European Championship campaign gets underway against Croatia at Wembley on June 13.
The centre-back’s absence from either would be a huge blow for club and country, but while Solskjaer concedes he will have to do without his captain for United’s remaining three league games, he hopes the 28-year-old will return in time for European action.
“The good news is it wasn’t broken, there was no fracture, but a ligament injury, of course, is serious as well,” the United boss said.
“If I’m very positive… that’s maybe stretching it, I’m hopeful he might be ready for it.
“I don’t think he’ll play in the league again before but we’ll do everything we can to get him ready for the final.
“You always try to look after players long-term. If he’s fit, he’ll play for us. If he’s not, he won’t.
“But, as I said, we’re hopeful that he’s ready and if he’s ready for us, he’ll probably be ready for the Euros. Norway’s not in the Euros, so I’m not really that concerned.”