England's Six Nations' hopes would be dealt a huge blow if Smith is ruled out, with fellow five-eighth - and former captain - Owen Farrell deemed ineligible for selection following his move last week from Saracens to French club Racing 92.
Farrell had already made himself unavailable for this season's Six Nations to focus on his mental well-being and Smith was a leading contender to fill his boots.
Smith left England's training camp on crutches and headed for a scan after his session was cut short.
"Hopefully it's very precautionary, but if he is not (available) it would definitely be a blow for us," England attack coach Richard Wigglesworth said.
"It wasn't a big incident. He was just jogging, but he pulled up and happened to be right next to a physio by the sideline, so they walked off after that.
"All the usual stuff (medical assessments) will happen."