Rashford has not featured for United since the FA Cup third-round replay against Wolves on January 15 due to a stress fracture in his back.
Originally, Rashford was expected to miss around six weeks and it was hoped he may return to action around April.
However, speaking prior to the Red Devils' Europa League clash at Club Brugge, Solskjaer said Rashford's injury was worse than first thought.
"He had a scan and it was maybe more severe than what we hoped and expected. He felt fine a few days before that," he said.
"I didn't know it takes that time to recover. When it heals he will be stronger for it so it's important we don't rush it.
"I would hope he's playing again this season it will just be touch and go towards the end.
"Hopefully we can get through this tournament [Europa League] and prolong the season.
"It takes time, I'm not a doctor but obviously I was hoping he'd recover quicker than it actually looks like he's going to be out for, another few months definitely."
England's first Euro 2020 fixture is against Croatia at Wembley on June 14 but, when asked if Rashford would be fit for the Three Lions, Solskjaer was non-committal.
"I would hope he plays before then but if he's not fit enough he won't go," he added.
United made a move to replace Rashford in the short term with the loan signing of Odion Ighalo in the January transfer window.
Solskjaer's Norwegian compatriot Erling Haaland was another tipped to move to Old Trafford before he opted to join Borussia Dortmund from Salzburg.
Haaland has proven an instant hit at Signal Iduna Park, scoring 11 goals in just seven appearances, and Solskjaer is delighted to see the 19-year-old impressing.
"I normally never comment but since I have had Erling I'm delighted for him," Solskjaer said of the striker he coached at Molde.
"He's a top boy I want the best for and he's started fantastically for Dortmund. For Norway it's brilliant we have a striker who can score again."