Please tell your direct supervisor as soon as possible as you already told your big boss and your big boss advised you to tell your direct supervisor.
Nothing can be worse if you tell you direct supervisor by yourself. Just tell him what has happened, what lesson you learned, and what action that you are taking to prevent it from happening again. Your supervisor would appreciate your attitude and honesty and your big boss would appreciate that you follow his advice.
In the worst scenario, if you tell your supervisor by yourself, they can't accuse that you are hiding your mistake (since your big boss knew it already and advised you to tell your supervisor) assuming that they want to make you in a trouble, right?
Just my two cents and good luck.
