A failing bearing in the waterpump will cause a 'rumble' or 'grumble' sort of noise. It's generally not intermittent though. It will be easiest to hear from outside with the hood up and will change with engine speed. The tough part is anything being turned by the belts could also be making that sort of noise (alternator, idler, air con pump, etc) so it's hard to isolate it. I have a mechanic's stethoscope that helps but isn't 100% The old-timer way to do it was to take a long screwdriver and place the end against the part you suspect and the hold the handle against your skull just behind the ear. The noise makes a vibration that gets transmitted up the screwdriver and directly to your ear in the same sort of way a stethoscope does, just not as well.
Additionally, any bearing can make that sort of noise when failing. Wheel bearings in particular are a common noise maker when they start to go. They are often a little louder when turning one way to the other since you load each side a bit more when turning. A wheel bearing will change noise with road speed and not make any difference with engine speed.
Another possible source could be a driveshaft. Yet another is a differential. Both of these would be road speed dependant instead of engine speed.
There's a lot of things it 'could' be, and without hearing it it's pretty much a guess. Even if I do hear it sometimes it's still a guess!
(Here's where I'd like to think doctors are more precise than mechanics, but from what I've heard from some of them they too are just taking an educated guess far more often than you'd like to believe )