Cole admits he rates Trafford ahead of Chelsea No1 Robert Sanchez.
Knowing Trafford is frustrated being regarded as second-choice at City behind Gigio Donnarumma, Cole told Paddy Power: "It's a difficult situation with Robert Sánchez. He makes good efforts sometimes, but he came out of the starting line-up the other day."
"This indicates that Liam Rosenior has not yet really decided who will be the first goalkeeper, even though he says he does not work that way."
Cole believes that Chelsea need a goalkeeper at the very highest level if the club is to be able to compete for titles.
"If a goalkeeper like James Trafford becomes available, I think Chelsea should consider making a move. Robert Sánchez is developing, but I think there is still a level to reach," Cole says.
"The question is whether he can reach that level in time for Chelsea to be able to challenge in the Champions League and compete with teams like Arsenal and Manchester City."