Rattling noise

It is possible that a loose pump can cause a rattling noise. It is more common for it to happen when the mounti barcket is loose. I just the pulley moves and the bracket is solid and you have tightened the bolts for the mounting bracket, the water pump needs replacement as it is failing and will probably go at the worst time possible, Muarphy's Law. If pulling on the water pump pulley either moves the bracket with it or tightening it solves the problems then pump is okay.
Check all the accesory mounting bracket for tightness, slack and check the tensioner for free motion,

The ohter cause s can be related to the A/C, so if turning it on or off affects the sound, unfortunately a lot of cars engage and disengage the clutch despite the A/C being on for system pressure and oiling purposes, so it will still be intermitant if the A/C compressor or clucth is the cause.
If you have not replace the timing belt in the last 60,000 miles, it needs to be replaced and that can cause rattling noises. If there is a performance change in the car it may have skipped a tooth or the crank/camshaft position sensor could be at fault.
Any loose part can cause rattling so it can often be a frustrating hunto for the issue. If you do all the above, get a long screwdriver and put the driver end to diferrent areas while you put your ear to the handle end. It makes a great sthethesope.
Listen to the top end as lifeters, valve clearance or worn camshafts can cause rattling noises too. You will usually hear more of a tapping noise but check it is noise does nto go away. Use the screwdriver to at least get into the area where the noise is coming from the give you an idea.
The water pump must be replaced thoguh and that might do the trick. I want to give you a diagnostic walk through link that meay be helpful.
Here is the link;

https://www.2carpros.com/articles/rattle-noise