Mario Balotelli's agent Mino Raiola has revealed that the striker will not be leaving Anfield this summer.
Graham Dorrans has signed a two-year deal with Norwich City following a successful loan spell with the club.
Conway is currently on loan at the Amex Stadium from Cardiff City and is enjoying a run of regular football which has helped him earn a recall to the Scotland squad.
However, Conway's loan spell is due to expire next month when he will have to return to Cardiff.
"It's going really well at Brighton. I'm playing, I'm enjoying it and that's what keeps me in with a chance of a Scotland jersey," he told The Argus.
"My emergency loan runs out on December 14, you're only allowed 93 days, so I can't go out on loan again or move permanently until January 1.
"I've got games to play before I need to worry about that. I could potentially go back to Brighton, though. I'll just need to go back to Cardiff and train.
"I'd rather be out playing than sitting rotting away at Cardiff. I'd hate to be one of those players who sits and just picks up his wages at a team. I want to play."