Former Chelsea and West Ham United striker Cole has been training with Celtic for a while now, but there is no contract in place at the moment.
Ex-Celtic striker Sutton has given his take on the situation and has asked why the Bhoys have not yet finalised a deal with the 31-year-old.
“Leigh Griffiths avoided a domestic ban this week but if he gets injured or suspended then Celtic are in big trouble as Nadir Ciftci’s not cutting it so far,” the former Chelsea striker wrote in The Daily Record.
“That’s why I can’t understand the hold up on Carlton Cole. He shouldn’t be here if there are fitness issues and if he’s fit then he should be signed up.
“Celtic can’t afford to wait until it’s too late and they have Griffiths injured and Ciftci’s on a ban.”
Celtic will return to action on Sunday afternoon, when they take on Hamilton Academical away from home at New Douglas Park in the Scottish Premiership.
The Hoops are second in the league table at the moment with 20 points from nine matches, four points behind leaders Aberdeen with a game in hand.
The Glasgow giants have won the league title in Scotland for the past four seasons, but this term they are facing genuine competition from Aberdeen.