The 21-year-old City midfielder, who spent a time on loan with the League One side last season, has turned down the chance to sign a permanent deal at Brunton Park, according to City's football administrator Brian Marwood.
"We accepted a bid for Adam Clayton from Carlisle but the lad wasn't too keen on a permanent move," Marwood told the News & Star.
"Adam enjoyed his time at Carlisle last season but he feels he could do a job in the Championship.
"Carlisle gave him a good grounding last year. Getting regular playing time in a competitive league has done him good.
"We would be happy to talk with Carlisle again but the final decision will come down to Adam.
"I think he would be an excellent signing for a Championship club but Adam could opt to stay and fight it out at Manchester City.
"It's down to him to show what he can do in training but we haven't ruled out either getting him back out on loan or on a permanent deal."