It's possible, because the water pump is designed to leak out of a weep hole behind the pulley when the bearing goes bad. If it's leaking that fast you should definitely be able to see it dripping or at least smell it under the hood. Are you able to spot any leaks?
Hopefully it's not a head gasket. I had the exact same thing happen to my car a few years ago (POS Ford 3.8 V6). I was losing coolant because it was seeping into the cylinders and being burned (you can often see this as white smoke out of the exhaust), and exhaust gases were also being blown into the cooling system which made it gurgle and bubble out of the overflow tank. Drive it around for a while then shut the engine off and see if the tank is bubbling, if so that's not a good sign.
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