A foundation star and current captain of the Giants, Greene is seriously considering ending his career back in his home state of Victoria.
Geelong, who have Greene's great mate and former Giants star Jeremy Cameron, are the logical destination should he depart GWS after 15 seasons.
"We're certainly hopeful that he re-signs with us and stays beyond this year," Kingsley told reporters on Friday.
"It's more than just Toby's potential last game.Â
"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."
In better news, promising young forward Max Gruzewski re-signed with the Giants this week until the end of 2028.
Gruzewski staying on could mean veteran Jesse Hogan seeks opportunities at a fourth club, just two years after a career-best season.
GWS have endured a rollercoaster campaign to only be the faintest outside chance of qualifying for the wildcard round.
Having destroyed reigning premiers Brisbane and beaten fellow grand finalists Geelong, the Giants have also, inconceivably, dropped games they should be winning.
GWS managed to lose to rebuilding West Coast when they met at Optus Stadium in May.
Another defeat to the Eagles on Sunday would ensure they miss the finals for the first time since 2022.
Engie Stadium will play host to a double-header, with the Giants' AFLW team opening their season with a match against West Coast after the men's game.
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