Rashford scored United’s second goal in a thrilling 3-2 win at Old Trafford but was replaced by Anthony Martial in the 86th minute.
Ole Gunnar Solskjaer said after the final whistle that Rashford would be assessed by United’s medical team after noticing some discomfort.
“Marcus felt his knee. We'll have to do a scan,” he said. "Hopefully it’s not too bad but I'm not a doctor, so we'll have to see tomorrow."
Rashford impressed in a more familar left-wing role, having switched to the right-hand side of United’s attack in recent weeks.
“Marcus has played up front, down the right and left,” Solskjaer said.
"We felt if we could switch the ball early today we could attack them because Liverpool defend on the front foot so we have to be quick on the ball. We did that well."
Solskjaer was delighted with the performance as well as the result.
Bruno Fernandes’ 78th-minute free-kick ultimately proved to be the difference in an end-to-end contest in which Liverpool struggled to contrain United’s quick counter-attacks.
“It’s a fantastic to win games over last year’s champions, a fantastic team,” Solskjaer said.
“We played on our terms, found a way the players believe in and we are getting stronger and stronger.”