The Lions fullback suffered a severe gash to his nose at Huddersfield in the Challenge Cup last Sunday.
Coach Steve McCormack said: “It’s quite nasty. He was told to go straight to hospital and may need some plastic surgery.
“He took a boot in the face and it’s cut right into the side of his nose. It’s a big blow.”
Mort is Swinton’s record points scorer in a season and McCormack will have to find someone to take over the goal-kicking role.
“We have a few options,” he said. “Chaz I’Anson can kick goals and so can Karl Ashall.
“Gavin Dodd is also a good goalkicker if I decide to bring him back. It won’t be a problem.”
Swinton gave new signing Chris Clarke his debut at Huddersfield.
Clarke, from Huddersfield amateur club Slaithwaite, has been training with the club since the start of the season.
“He did really well and I was impressed with him,” said McCormack.
“He had about 25 minutes and made a good impact when he came on.
“It’s a good thing to have another player joining the ranks and one I know I can trust in the big games.
“It does give me a selection headache but that’s what we want.”