Manager Kavanagh has been in touch with the Sheffield Wednesday player during his five-month term in jail.
"I sent him letters to keep his spirits up and if Wednesday decide he's not their cup of tea, I'd take him without a shadow of a doubt," Kavanagh told BBC Radio Cumbria.
"I'd hoped to visit Gary and was in contact with his uncle to try and sort it but I didn't manage to get over.
"He's still under contract, but I think there is a real possibility that Sheffield Wednesday could let him go.
"There's a new manager now that may want to look at him first, but certainly I would be interested in signing him."
Wednesday refused to comment on the 23-year-old's future following his release.
Madine was sentenced to 18 months in jail for two assaults, one causing grievous bodily harm and the other actual bodily harm.
The forward, who scored four goals in 35 games last season, tweeted that he was looking forward to return to training.
Madine played for Carlisle between 2007 and 2011 and trained with the club prior to his sentence.
Graham Kavanagh must think he is some kind of rehab magic man or something, he's already signed Courtney Meppen Walter after his prison release and now he's going for the boy Madine.- dannyjohnson