It's probably the "pluck" produced by the piezo transducers (I know it by heart, I have a Variax 500 since 2003). Some FX's and amps enhance it. Playing through Line6 modeling diminishes it. There's no way to cancel it totally since it's due to the hardware used. The design of the JTV is supposed to have solved this issue.

Possible tricks against it:

1) try a pre-distorsion EQing set around 1khz and scoop the mids drastically with the highest Q factor available;

2) plug the axe in Line6 devices only;

3) adapt your playing: avoid dead notes and palm muting (that's what I've refused to do);

4) don't use too much gain with Variax modeling (that's what I tend to do);

5) don't focus on this issue: even under high gain, it disappears in the mix once the guitar is played on stage - that's why my Vax is still my main tool for live gigs after almost 9 years of use. :-)