After a while into my weightlifting sessions at the gym, I'm noticing in the upper arm area a jerking/snapping/movement of some kind. The only way I can describe it is that something long (maybe an artery or vein, or skinny muscle fiber) "snaps" in and out of place.

Doing bicep curls I feel it in the area between the bicep and the inner elbow. Doing knuckle pushups while engaging the triceps, I feel it in almost the same area, only on the tricep side. This only happens once my muscles have become fatigued and stiffer than normal. It's not painful per se, but it's bothersome and it worries me a bit.

Whatever it is it doesn't feel right, and this doesn't happen on my left arm. Any ideas?