The striker was absent from the squad for Monday's (AEDT) Premier League trip to Liverpool after failing to recover from the problem that forced him off during the 1-0 FA Cup third-round replay at Old Trafford.
There were some suggestions the injury was more serious than first feared, although Solskjaer insisted this week the club would do everything to help Rashford be ready for the match at Anfield.
However, speaking prior to the game, Solskjaer revealed United are likely to be without their leading goalscorer until after their mid-season break in February.
"He got a couple of knocks again, jolts when he came on against Wolverhampton, and he's aggravated his back," Solskjaer told Sky Sports.
"He's had some trouble before. We're going to give him time to recover and rest before we put him back on the pitch.
"He's always recovered quickly before, so let's hope he does that again.
"I wouldn't expect him to be back in the next few weeks. We've got a mid-season break, but I'm not sure we're going to see him before then."
After the meeting with Liverpool, United host Burnley in the league on Wednesday before an FA Cup trip to either Watford or Tranmere Rovers next Sunday.
It then faces Manchester City away in the second leg of the Carabao Cup semi-final, trailing 3-1 from the first game at Old Trafford.
United faces Wolves at home on 2 February (AEDT) and then expects to have 16 days off before heading to Chelsea, with the first leg of its UEFA Europa League last-32 tie with Club Brugge taking place three days later.