Outer Forearm Pain while Pushing

I started to have unbearable outer forearm pain while doing bench presses especially. It starts on the outer forearm, close to the wrist and radiates till about 2 inches above the elbow. There is no pain while doing deadlifts or other back work, but pushing work like presses especially bench hurts it quite a lot.

I started using some good wrist wraps; it alleviates the pain, but its still there. I don’t do any isolation work for my arms except for some laterals. The pain goes away if I stop rep work with bench/close grip presses, but gets back instantly as i start them.

Is it because my wrist is bending too far back while pressing, should I start using a thumbless grip to ensure that the bar is directly over my wrist?