Green has missed the Warriors' last two games after suffering an ankle injury in the Challenge Cup win at his former club Hull KR earlier this month.
The 26-year-old had hoped to be back to face Widnes in Super League on June 7 but Wane says that is not realistic.
"He is the eternal optimist, Greeny. He'll be wanting to play," said Wane. "But when he got injured he said he'd be back in two weeks and we're now three weeks on.
"I like that, he's keen to get back, but I'm more of a realist. I'm not going to rush him. Once he is 100 per cent ready to play I'll give him another week.
"It is an awkward injury - it is hard to rehab and hard to treat.
"He should be okay in two or three weeks' time."
Warriors captain Sean O'Loughlin and hooker Michael McIlorum are both doubtful for Sunday's Super League trip to Wakefield.
The pair will be assessed after suffering knee injuries in the Magic Weekend defeat of Leeds.
Youngsters Greg Burke and Scott Taylor will be included in the squad.