Grandison has been managing the complaint over the last couple of weeks but was eventually forced off at half-time in Saturday's defeat at Swindon.
A leading hip specialist in London has now told him to rest in the short-term before a decision is made over the next course of action, with an operation not ruled out if the injury fails to improve.
Manager Graham Turner told Town's official website: "The diagnosis at the moment is he should have two weeks rest away from training and if it settles down they will inject it. But give it two weeks rest and see how it goes from there.
"The worst-case scenario is he'll need surgery which could keep him out for months rather than weeks, but we're hoping it will settle down with some rest. We've got the best people looking at him so it remains to be seen how long he's going to be out."
Turner continued: "It's a possibility it will be a month and then a return to training and it's then probably another two weeks at least getting him fully match fit.
"I think the best-case scenario we're looking at is probably six weeks before he's fit but it's a problem he's got to manage now and we'll see over the next few weeks. I think that will be crucial."
And Turner has not ruled out turning to the transfer market to bring in another defender if needs be.
"If we lose somebody of Jermaine's stature there's thoughts that we need to bring somebody in," he added.