Rashford came on in United's 1-0 FA Cup third-round replay win over Wolves at Old Trafford, but he lasted only 16 minutes before being replaced having sustained a blow to the back following a challenge from Matt Doherty on Wednesday.
Though Rashford was on the field when Juan Mata scored the winner, Solskjaer conceded the decision to bring him on "backfired" as he sustained an injury four days before the trip to Anfield.
Rashford has scored 14 goals for United in the Premier League this term and Solskjaer said the club would do all they could to get the 22-year-old fit for the weekend.
"He's a quick healer and he'll play through some pain, definitely, if he's capable, if it's not a bad, bad injury," Solskjaer told reporters at a news conference.
"But we've seen that he's struggled lately - but more when he's been tired.
"He wasn't on long enough to be tired so I think it might have been something that just happened there and then.
"He's been absolutely top this season so we'll do everything we can to get him fit for Sunday.
"If he's not then we'll play without him."
Rashford himself tweeted after the match to reveal he attempted to carry on, while congratulating his team-mates on booking a fourth-round tie with either Watford or Tranmere Rovers.
United were already without key midfield duo Scott McTominay and Paul Pogba, as well as defenders Eric Bailly and Luke Shaw on Wednesday, yet Solskjaer was not about to bemoan an increasing casualty list.
"It's football, I'll never ever complain about all the injuries we've got," he added.
"That's just part and parcel of it and at the moment that's Marcus. He's been available and fit all season so you can't complain."
Liverpool are storming towards a maiden Premier League title having taken 61 points from a possible 63. The only points they dropped came at Old Trafford earlier in the campaign when Rashford scored in a 1-1 draw in October.
Solskjaer is therefore confident his team will have a game plan to make life difficult for Jurgen Klopp's Reds.
"Of course it's hard but I've had Liverpool here twice and we've drawn both," said the Norwegian.
"Of course we want to win those but I think we've found a nice way of playing against them and we've deserved the points in both of them."