Gibson has been struggling with the thigh problem for some time, but Everton have been able to manage the injury of the Republic of Ireland international.
Moyes admits Gibson's injury could see him break down at any time, but the Everton boss has hailed the former Manchester United man for playing on with the problem.
"I think we are in control of Gibson but there is a danger," said Moyes.
"We thought he might need an operation earlier in the year but the medical staff have managed it well. We think we can get away with it just now but he will need an operation.
"I don't question his character as he came from a club that instils that in you.
"We're not going to let him roll over so we can tickle his belly.
"He's going to have to get up and get on with it - and he has done, to be fair."