The Giants are 12th with an 8-11 record, needing wins in their four remaining games to potentially pinch a berth in the expanded 10-team finals format.
Midfielder Callaghan (ankle) will miss the rest of the season but Greene (quad) could return next week.
Ruckman Kieren Briggs (leg) is likely to return on Saturday to face Port Adelaide on the road, while coach Adam Kingsley expects both Jake Riccardi (ankle) and forward Ollie Hannaford (foot) to come into calculations.
"Toby ran today without pain, which is great," the coach said on Thursday.
"There's a feeling he might be okay to go next week ... it's looking okay. We'll assess him this time next week."
"I haven't caught up with our medicos yet but imagine he's (Briggs) going to be fine and (Riccardi and Hannaford) whether they play AFL ... maybe they need a run in the VFL, but we'll discuss that."
Kingsley said there would be no white flag raised while they were still in the finals hunt.
"It's still right there in front of us if we're good enough," he said.
"Port (15th, 6-13) have been really good; the ladder doesn't suggest how good they've been.
"They're defending well, their stoppage game is in good order. They're going to throw some punches.
"We have to make sure we manage those and throw a few of our own."
Kingsley's side have dismantled two-time defending premiers Brisbane, beaten Geelong and knocked off runaway ladder leaders Fremantle this season.
But they have also lost to bottom-placed Essendon and the 16th-ranked West Coast.
They trailed Sydney by 53 points at halftime on Saturday and eventually lost by 51.
"We didn't compete, full stop, really," the coach said after reviewing the Swans loss.
"Sometimes we've been pretty good, sometimes we haven't.
"Becoming more consistent is something we've spoken about a little too regularly."
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