The 21-year-old failed to report for the Garry Lynch Memorial friendly between Junior side Cambuslang Rangers and a Light Blues reserve team on August 11.
The no-show prompted an angry response from boss Ally McCoist, who has told the player to keep his distance from the club’s Murray Park training complex for the time being.
Cole has yet to make a competitive appearance this season and his last outing was in a friendly against Elgin City in July.
McCoist told the club’s website: “Darren failed to report for the game against Cambuslang but he’s not following a fixed suspension.
“He didn’t attempt to contact anybody at all to let them know he wouldn’t be coming to the game and it goes without saying we thought his conduct was totally unacceptable.
“I’ve asked Darren to stay away from the club until further notice and until we discuss what our next course of action will be.
“I’ve already had a chat with the player to register my disappointment with him and to ask him for an explanation.
“What was said there will remain between him and I but I’ll also need to have a talk with my staff to decide what we will do now.”