Rashford was in flying form before the 22-year-old's picked up a serious back injury during Manchester United's FA Cup replay against Wolves - and that is not only a blow for United's top four hopes but also England's Euro 2020 campaign.
Solskjaer expected Rashford to be out for six weeks, but the United manager admits the forward could miss the rest of the season - and perhaps even Euro 2020.
"I would hope he's playing this season but it'll just be touch and go towards the end," the Norwegian said.
"So hopefully we can go through and get through this tournament and prolong the season.
"It takes time. I'm not a doctor, but obviously I was hoping that he would recover quicker than it actually looks like he's going to be out for. So, another few months definitely."
Euro 2020 doubt for Rashford
Asked whether he would be in jeopardy of missing the Euros in that case, Solskjaer said: "Well, I would hope that he plays before then but we're not sure, so if he's not fit enough he won't go."
With skipper Harry Kane also lied low, Solskjaer's injury update will be the last thing England manager Gareth Southgate would want to hear.
Rashford, who came off with a back injury against Norwich just days before the Wolves match, had been confident of being "back before the season ends" shortly after sustaining the injury last month.
Asked if there had been a setback, Solskjaer said: "He had a scan and it was maybe more severe, that fracture, than what we hoped for and expected because he felt fine the few days before that.
"I've never had anatomy (classes) in school so I didn't know it just takes that time to recover.
"But when he recovers, when he's healed, he'll be stronger for it, so it's important that we don't rush him. And we won't do that."
United had already looked light in attack before Rashford's injury, with deadline-day loan signing Ighalo the only fresh addition.
The Nigeria striker has travelled to Belgium with the squad, but McTominay stayed at home despite training with the squad on the eve of the match.
Paul Pogba is another high-profile absentee in Belgium, with the recovery from an ankle injury continuing slowly to background noise about his future.
"He's not been part of team training yet, so it'll depend on how he feels or when he feels ready to do that," Solskjaer said. "He's not anywhere near that yet."
IMPORTANT: Sporting Life podcasts have new homes across the below platforms.
Please make sure you subscribe to ensure you don't miss future episodes of your favourite show.
Follow Sporting Life on social media - find us on Facebook here or tweet @SportingLifeFC
Related football links
