The Giants banked a precious 12-point away win over Port Adelaide on Saturday night to rekindle hopes of reaching the finals.
They have three games remaining - against Gold Coast, West Coast and Adelaide.
The Blues hold 10th spot with 40 premiership points after their upset thrashing of reigning premiers Brisbane on Saturday night.
St Kilda (36 points) are 11th, ahead of GWS (36 points) on percentage.
Inspirational skipper Greene has missed the past two games because of a thigh injury but could return against the Suns.
"He'll need to get through pain-free and train well next week, fully train, in order to be available," coach Adam Kingsley said.
"Whether he can do that, I'm not sure. It will be touch and go, I imagine."
Kingsley said in the race for a wildcard berth, the Giants are embracing the 'never surrender' slogan introduced by inaugural coach Kevin Sheedy.
"We'll continue to fight," he said.
"One of our mantras is never surrender.
"And I think we live by that, whether it's in games, playing to the last minute, or whether it's in the season, you'll see that out of us."
Kingsley took heart from his side still being in the reckoning for the 10th spot despite a spate of injuries this season.
"If we're good enough, we'll make it," he said.
"If we're not, we won't - it's as simple as that.
"We're competing with Carlton and St Kilda for that last spot in that wildcard round and it will be the same story for those guys too."
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