The centre picked up the injury in the first half of the 34-6 victory over Ospreys, and was replaced in the second half despite having the hand strapped.
Director of rugby Jim Mallinder told BBC Sport: "Fingers crossed there's no permanent damage and no broken bones. He gave it a chance in the second half but just couldn't quite get that grip going.
"We're going to have to see how that is. He'll probably have to go for an X-ray."
Burrell, who has seven caps for England, could miss the upcoming Autumn international if he has suffered a fracture.