That's probably what you have. You have basically 2 options, as far as I know. Water around them and enjoy them, or toss out the nests as they are built untill the birds get sick of you. You might try putting little rubber snakes in the baskets, but I'm not sure if it would work or not.
House finches are sweet.
House Wrens nest in cavities and would leave your plants alone. This happened to me and I would remove the nest every time they would begin to build and eventually they got tired after 3 or 4 removals. We had been watering lots before we realized there were babies in there. They probably knew how to swim before they knew how to fly. They made it through.