Rashford suffered a double stress fracture in his back against Wolves last month, with a six week prognosis given after the January 16 clash clash.
But Solskjaer says the England international will be lucky to play again this season, and could even miss the European Championships in June and July.
"I would hope he can play again this season. It'll just be touch and go towards the end. Hopefully we can get through [the Europa League] and prolong the season," Solskjaer said.
"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]."
He added: "He had a scan and it was maybe more severe than what we hoped and expected. He felt fine a few days before that.
"I didn't know it takes that long to recover. When it heals he will be stronger for it so it's important we don't rush it."