I came across this post when trying to find an answer for my issue - a high pitched buzzing/whirring sound when my Whirlpool Estate Washing machine was agitating or spinning, no noise when it was filling with water. It continued to operate fine, it was just loud. Fearing something might start on fire if I continued to use it, I hired a repair person because I couldn’t lift the cabinet off the washer to try to figure it out myself using the checklist posted here. I’m going to share what was found in case someone else has this type of issue. After removing the cabinet, you could hear where the noise was coming from, under the tub. He disconnected the water pump to see if that was culprit. Noise stopped. Problem isolated on the first try. When he removed the pump , there were not one but two underwires in there. No garments with underwires were washed since the summer (bathing suits) - and after a thorough check of everyone’s wardrobes, none were missing. So, not sure how long they were in there before they made there way down to the pump. He replaced the pump because the wires had done some damage in there. As long as you are pretty strong, or have someone to help you remove the cabinet from the washer, this was a pretty easy fix.
Honestly, during my search I read that my problem could have been something stuck in the water pump, or the motor, or a bearing, and I thought the water pump scenario was the least likely - what could have possibly been stuck? We always check pockets, and no buttons or zippers, etc. were missing. I am happy I was wrong because I think the pump might have been the cheapest of the 3 scenarios. Hope this info helps someone else!