During the FA Cup's third-round replay that Manchester United ousted Wolves 2-1, Rashford was sent on the pitch by Solskjaer as a second-half substitute but had to be replaced after 16 minutes.
"He got a couple of knocks and jolts when he came on against Wolves and aggravated his back," Ole was quoted as saying by the club's official website. "It's a stress fracture and it's not something that's happened before. It's so unlucky for Marcus, he'll be out for a while.
"He's had some trouble before and we're going to give him time to recover and rest before we put him back on the pitch. He won't be back until after the mid-season break. I don't know long it is, I'm not a doctor, but normally a stress fracture is six weeks to heal and then he needs some recovery or some rehab after that probably.
"He's always recovered quickly before so let's hope he does that again. This time it takes longer so we just have to keep monitoring him, do another scan over the weekend and see how he is. We're going to give him the time he needs," Ole added.
The 22-year-old has scored 14 goals so far this season and is the team's top scorer and third in the Premier League.
Manchester United lost to league leaders Liverpool 1-2 on Sunday without Rashford and are at the fifth place with 34 points, five points away from fourth-placed Chelsea.
( With inputs from IANS )