Rashford, United's top scorer this season with 19 goals, is a major doubt for Sunday's Anfield clash after suffering a back injury in a 1-0 FA Cup win over Wolves .
The Mirror says United do not want to rush him back too soon and risk the 22-year-old suffering a further injury that could ultimately rule him out for even longer.
But Rashford is said to have responded well to initial treatment and is ready to put his body on the line and play on Sunday – even if he requires a pain-killing injection to do so.
On Rashford's injury, Solskjaer said: “He's had trouble with the back lately but that is a different matter. Now we've got a different one.
“We'll do everything we can to see and try to get him fit for Sunday, but I can't promise. We'll manage."