It’s understood the Nigeria-born ex-England striker is attracting strong interest from the Middle East, and he is keen to see if that results in an offer.
Parkhead chief executive Peter Lawwell held talks with Cole and his agent after the 31-year-old impressed boss Ronny Deila on trial last week.
The former West Ham and Chelsea hitman is considering his options and is aware there could also be renewed interest from England.
However, Celtic first-team coach John Kennedy is still hopeful the free agent can be persuaded to come to Scotland.
He said: “Carlton is fit enough but he hasn’t had football training or any games this season.
“It was difficult to judge him properly. It’s now been left with the manager, Peter Lawwell and Carlton’s representatives.
“The manager spoke to him before he left and said he’d be in touch. Peter was to speak to his representatives to see what the deal was going to be.
“It obviously has to be something that is good for both parties if a deal is to be done.”
