Rashford, 22, was replaced in the 80th minute after he had been on the pitch for 16 minutes as United beat Wolves 1-0 in their FA Cup third-round replay.
The England international has scored 22 goals for club and country this season in what is already his most prolific campaign yet.
United have arguably their biggest game of the season at the weekend. Going to Anfield without their talisman and getting a result could be a tall order.
Speaking to the press, OGS said “He’ll be tested in the next few days – we have to see for Sunday,”
“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.”
Asked if the decision to play Rashford backfired, Solskjaer replied: “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.”
In the cup, United will be playing either Watford or League One side Tranmere away from home in the fourth round of the FA Cup on 26 January. The League is still serious business for the Red Devils if they look to topple Frank Lampards Chelsea for a space in the top 4