Solskjaer intended to keep Rashford on the bench during Wednesday’s third-round replay at Old Trafford, but United’s listlessness up front led to the 22-year-old England forward coming on as a substitute in the 64th minute.
Moments later, Juan Mata burst clear and chipped over John Ruddy to hand United a 1-0 win, but Rashford’s cameo ended after 16 minutes, shortly after being discomforted by a challenge from behind by Matt Doherty.
He clutched his back as he left the field before heading straight for the tunnel, and United now face an anxious wait to discover whether their 19-goal top-scorer will be fit to face the runaway Premier League leaders.
Solskjaer insists United will do all within their power to get Rashford available for the Anfield trip on Sunday afternoon.
The United manager said: “I didn’t want to play him. I think he got a knock, a knee or something, in his back, but he’s been struggling for a little while.
“That’s why we’ve kept him away (from starting against Wolves), but we needed the win. He was involved in the goal so that was a positive but it backfired.
“He’s a quick healer and he’ll play through some pain, definitely, if he’s capable and it’s not a bad, bad injury, but we’ve seen that he’s struggled lately.
“But that’s been more when he’s been tired. He wasn’t on long enough to be tired so it might be something that just happened there and then. He’s been absolutely top this season so we’ll do anything and everything we can to get him fit for Sunday. If he’s not then we’ll play without him.”
Rashford expressed his frustration at having to be substituted, writing on Twitter: “Never want to feel like I’ve let my team-mates, the club and most importantly the fans down so did my very best to carry on tonight and was gutted to come off.
“Top work from the lads to get into the next round.”
United, who will next face either Watford or Tranmere in the fourth round later this month, are without a number of first-team players because of injury, including midfield duo Paul Pogba and Scott McTominay.
But Solskjaer added: “It’s football. I’d never, ever complain about all the injuries we’ve got, that’s just part and parcel of it. Marcus has been available and fit all season so he can’t complain.”
Wolves were left frustrated by the video assistant referee as Pedro Neto’s early strike was ruled out.
Raul Jimenez’s unwitting handball in a challenge with Harry Maguire meant Neto’s finish counted for nothing.