The 21-year-old made 36 appearances and scored three goals for the Cumbrians during his successful loan spell last season.
Clayton has attracted interest from Championship clubs but Abbott believes the player is within Carlisle's price range.
He told the News & Star: "I have an idea what it would cost us and that would be okay. First I need to speak to Manchester City to find out what their thoughts are. Then I will speak to his agent and then the player himself.
"Adam is keen, but he will look at his options. He has done well enough here to have attracted attention from elsewhere. He has got to be playing every week and that might stand in our favour, as he might not play every week at a Championship club.
"We have to give Manchester City massive respect. They might see him as part of their first-team plans. If not, we have to see where we lie in the pecking order. If they gave us the go-ahead I would do it this afternoon."