The United striker is due to have surgery, which is expected to take up to three months to recover from.
Rashford had been complaining of issues with his back before taking a heavy blow to it in United's FA Cup third round replay with Wolves.
He has now suffered a double stress fracture that will require non-invasive surgery.
Solskjaer said later on Sunday: "It's a stress fracture which happened against Wolverhampton. That is usually six weeks and then he will need some rehab after that."