The Welshman has been out of action since April, but was nearing a first-team return after playing in three recent reserve team games.
The 25-year-old injured his knee in training on Monday, however, with subsequent MRI scans confirming he may need further surgery to repair the problem.
Manager Stale Solbakken told the club's official website: "Obviously, this is the worst thing that could have happened to Wayne.
"It is a tremendous blow to Wayne and the club, but we will do everything we can to help him through this."
Hennessey said: "It was just one of those freak things that can happen and it's no use dwelling on it - nothing is going to change things.
"So I'm just concentrating on getting fit again - I've done it once and I'll do it again."