The thing that the water pump bolts onto

Thanks for the advice, AJ. I’ll wipe it clean and watch it again. It sure seemed like it was weeping out of the metal. I know that sounds hard to believe, but I wondered if Father Time had eroded pin holes right through the metal
You can go to your friendly auto parts store and they will lone you out a cooling system pressure tester. That way you can pressure up the system when the engine is cold, then you can use your hands to find the leak without getting burned.