The trio - all security personnel at Stamford Bridge - have been suspended from their jobs pending a full investigation.
Reports claim that another member of staff noticed that one of the handles was hanging off the side of the famous cup, and reported the damage to senior club officials.
The trophy was immediately sent to a London silversmith for emergency repairs, and is now back as good as new - but furious club chief executive Ron Gourlay has ordered a thorough internal enquiry into what happened.
The Daily Mail reports that the three security staff were not on duty at the time of the incident, and were in the building without permission.
A club spokesman confirmed that the staff members had been taking pictures of themselves with the trophy when the accident happened.
Trophies at Stamford Bridge are usually kept in a locked cabinet, but the cup - won with a famous penalty shoot-out triumph over Bayern Munich in the final - was on open display as a reward to the club's staff.