GWS coach Adam Kingsley has acknowledged Sunday's home game against West Coast could be "Toby's last potential game" in front of a Giants crowd.
Greene is weighing up his future, with Geelong believed to be interested in pairing him with former teammate and best friend Jeremy Cameron.
“We're certainly hopeful that he re-signs with us and stays beyond this year,” Kingsley said. “So that's where we're thinking.
“As he's said all season, he'll get to the end of the year and then have the discussions and make a decision. We're hopeful, but we want to win. It's more than just Toby's potential last game. It's giving the fans some joy and ensuring that we finish off the season really strongly.
“With that faint hope of Wildcard Round, we can give ourselves every chance to be able to participate in that. That looks unlikely, but we still want to perform really well.”
Uncertainty doesn't just reign over Greene's future, but also over veteran forwards Jesse Hogan and Jake Stringer, who are both unsigned beyond this year.
Max Gruzewski's two-year extension, despite the club tabling a four-year deal in front of him, makes an intriguing off-season for the Giants.
“Very happy with his signature,” Kingsley said.
“He's becoming quite an important player for us; he's done a couple of really good jobs, he's competed well ahead of the ball, and he's finishing his work pretty well.
“He's only young still. So, that key forward grouping is quite promising for us, and beyond this year. There's growth in all these guys; we've seen that with Cads over the year. I think there's another couple of levels that Max can go to.
“It's possible (Hogan remains at the Giants). But I mean, we'll have those discussions at the end of the season, and that'll be up to Jesse and his management, I think.”
