They're both living organism, just like us. Please articulate your reasons for why you believe they do or do not possess souls.
I honestly think they do have souls. I have a pit bull and an African Grey Parrot. I watch the pit when he sleeps sometimes and I often notice, that he runs in his sleep, he barks, he smiles and he pants, all while sleeping. Some of you may or may not feel different about what I am about to say but my personal belief is that when you sleep, your soul travels. The physical rests because it's low on energy, the soul is energy and doesn't need to rest, so when the body rests, the soul travels. So, under that thought, if my dog is doing all that activity in his sleep, then he must be dreaming about something that he is responding to. If he is dreaming, he is travelling, he is physically in my house, so it must be the soul.