It's more to do with the "grinding" action on your wrists from having them locked in position (not able to move at different parts of the action...not having a more neutral/natural grip).
You'd be surprised at the amount of exercises that contribute to this issue (not just the exercise that causes it to show). For me, I have to use the EZ bar for curls. Same can happen with pullups when load gets really heavy.
Apart from wraps and hand possition, I'd suggest looking at your other exercises as well, and if not sorted, switch to dumbbells for 2-3 months and then go back to the barbell.
BTW, my brother has more issue with this than me and the problem is likely over-use of the wrist joint (insert joke here) rather than them being weak, because he loved (READ obsessed over) forearm/grip strength and as he progressed more and more on it, he had to back off because it really aggravated his wrists while benching.