Rashford has not turned out for the Red Devils since January 15 due to a back injury, although it had initially been thought that the attacker would return to action towards the end of March.
Solskjaer, though, speaking at his pre-match press conference ahead of Thursday's clash with Club Brugge, has claimed that the 22-year-old could miss the rest of the season and also Euro 2020.
"I would hope he's playing this season. It will just be touch and go towards the end. Hopefully we can get through this tournament and prolong the season," the United boss told reporters.
"It takes time, I'm not a doctor. I was hoping he would recover quicker than what it looks like he's going to be out for. Another few months definitely. If he's not fit enough he won't go [to the Euros]."
Rashford had been in excellent form for the 20-time English champions in the first half of the campaign, scoring 19 times in 31 appearances in all competitions.
England will open their European Championship campaign with a clash against Croatia on June 14.