Grumbling noise before/after gas motor runs

Had wrote about noises happening before and immediately after gas motor starts and stops. I was under the impression the coolant pump came on about 1 minute after shutdown. My noise happens within 5 to 10 seconds before and after gas motor running and stopping. After posting, I went to reading more and more, and now it seems this coolant pump may also be my problem, unless there is more than one electric motor involved? I still get the other pump noise, just fine, approximately 1-minute after car is off. Is there another electric pump motor? What do I tell the mechanic to check? Thanks

Wonder, it is Federal Law that emission components be covered under warranty for 70k miles. Also saw this in Toyota handbook. This coolant transfer pump could be part of the low emissions system, right? Anyone try arguing this to Dealership, or to Toyota?

And one more question? Assuming it is a bearing going bad, can it be oiled or greased?