Hartley missed the 22-16 loss to France in Paris on Saturday - a defeat that crowned Ireland Six Nations champions - with a calf strain and the skipper is still to be given the green light ahead of the upcoming clash against Ireland this weekend.
The 31-year-old hooker was able to run on Monday but Jones, who has seen his side lose consecutive games for the first time since taking over in November 2015, says more information will be provided on Tuesday.
"We will find out a bit more tomorrow," he told Sky Sports. "He needs to run a bit faster tomorrow. We still don't know whether he will be available or not. If he is fit he will come straight back into the team."
England are almost certain to be without winger Elliot Daly after he injured his foot against Les Bleus and the Red Rose are currently awaiting the results of a scan to find out the extent of the issue.
Mike Brown, who was dropped to the bench for the France loss, is expected to return to full-back, with Anthony Watson going back to the wing.