The club said it was taking "appropriate action" over the matter.
The News of the World reported that Cole did not know a .22 air rifle was loaded when he fired it at 21-year-old Tom Cowan.
It was reported that the sports sciences student, who is on a one-year work placement at Chelsea, was hit in the side by a lead pellet at the training ground in Cobham.
Chelsea's medical staff reportedly treated the bleeding wound and Cowan did not require hospital treatment.
A club statement read: "We have fully investigated the incident and we are taking appropriate action. We will not be commenting further as it is an internal matter."
A Surrey police spokesman said: "We can confirm that while no direct allegation has been made to Surrey police, the matter has been brought to our attention through reports in the media, and we will be contacting the club in due course to establish whether any criminal offence has been committed."
Source: PA
Source: PA