Remove the fan belt and see if it goes away. Water pump failure usually makes a whirling or grinding noise. Of course while the belt is off you can check the water pump for smooth operation. It should turn smoothly with a minimal amount of slop. Look at the 'weep' hole below the impeller shaft for signs of coolant.
There is an issue with the cam sprocket and a short or long dowel causing wobble and premature chain failure. Just some thing to keep in mind.
Get an engine stethoscope. They are cheap and work so much better than a screw driver (you can imagine how I know that
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Doesn't sound like a water pump from your description but at least you will be able to eleminate it as a possibility.
Rick