Salmon, 19, made three substitute appearances for the Cumbrians, making his debut at Bournemouth at the end of last season before joining Alloa on loan in August, while Gwinnutt, 18, did not appear for the first team.
Both players progressed through the club's youth system.
"They have worked hard and they've gained some good experience with the loan spells they've had," manager Graham Kavanagh told the club's official website.
"I think it goes without saying that we wish them every success as they move on from the club."
- Sports & Recreation