Young forward Logan McDonald will also be out of action for about two months after he suffered an ankle injury in Saturday's AFL home loss to Fremantle.
Last year's grand finalists are two games outside the top eight with a 3-6 record and already have Dan Rampe (neck), Paddy McCartin (head) and Sam Reid (hamstring) out of action.
Mills was hurt early in Saturday's match and scans have confirmed a moderate strain.
McDonald did not return after half-time following an injury sustained in a marking contest in the second quarter.
The Swans say he has a significant medial ankle sprain.
"Callum is obviously one of our on-field generals, and Logan has been building into some good form, so we are disappointed to lose them for what looks like an extended period," said Sydney football boss Charlie Gardner.
Geelong forward Ollie Henry is out of action for a couple of games after he suffered a ruptured testicle late in the loss to Richmond.
His brother Jack could be back for the first time this season after recovering from a foot injury, while Tyson Stengle (arm) is also in the mix for Saturday's game against Fremantle in Perth.
"Late in the game Ollie Henry had a testicular rupture from a collision, so post game he reported it to the club, to our medicos, and he's had that repaired so he'll miss a couple of weeks," said Geelong football manager Simon Lloyd.
Also on Tuesday, North Melbourne said star onballer Luke Davies-Uniacke could be out of action for five weeks because of the hamstring injury he suffered in the weekend's loss to Port Adelaide.
Forward Jack Mahony needs surgery after hurting a shoulder against the Power, ending his season.
Cameron Zurhaar (cheek) is out for their next match against Sydney, while Darcy Tucker (hamstring) will be tested.
Flynn Perez and Lachie Young are also sidelined under the AFL's concussion protocol.
Meanwhile, St Kilda player Zak Jones has suffered another setback and is now sidelined with a right quad muscle strain.
Jones has not played this season because of an achilles injury and a scan on Monday confirmed this new issue.
He was due to return through the VFL in the next fortnight, but now is out of action for six weeks.
"We're really feeling for Zak, who has done everything possible during his injury rehab," St Kilda football manager David Misson said.
Jones's setback headlines mixed injury news for the Saints, who might regain key forward Max King for Sunday's away game against GWS.
While King could be a big inclusion, fellow forward Tim Membrey is out because of concussion.
But small forward Jade Gresham should return after he missed the loss to Adelaide with a knee injury.
Jimmy Webster (cheekbone) made a successful return through the VFL, but Jack Steele is in some doubt with a medial ligament strain.
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