What I think we need to do here is establish what kind of click we're dealing with here.
1) Cut your fingernails short on your plucking fingers, otherwise it'll sound like you're playing with a guitar pick. I have mine shorter than half a millimeter, so you might not be cutting them short enough. Does this help?
2) If not, try pushing down the strings until they hit the magnets on the pickups. This should produce quite an audible click, if it's the same as the one you get, try rising the strings or lowering the pickups
3) If that doesn't help, try slamming down the strings on the fretboard with either hand, doesn't matter, without muting the strings. You want it to be like aggressive fretting, not clean tapping. Does this resemble the click you get? If so, try adjusting your action or your playing technique, starting with the latter one as string action usually doesn't cause clicks but buzzing. I'd look closely at how you fret the notes, as that can cause audible clicks.
4) Try tapping the strings with your right hand while muting with the left. This is the kind of unconscious tapping that annoys drummers, usually if you're aware of the problem it goes away, but you might wanna keep it in mind.
5) Now, if non of these steps helps, you have to start looking at odd things like loose tuning keys (they usually rattles more than clicks but who knows), bridge, etc. Also have a look at the electronics, you can get pretty loud clicks that mutes the sound of the bass if you have a loose solder joint somewhere, or maybe a faulty pickup (try tapping the covers with the amp on, if it produces anything but a clean tap sound you should see to get them fixed, esp. if they crackle and pop), bass tones can cause all kinds of vibrations.
If you don't find it this way, take it to a tech and play a bit for him so he can hear the sound in person, that should help him find it a lot faster.