Years ago, a Sunday School teacher that husband and I had was adamant that animals did NOT have souls. So strongly were her beliefs that she wanted her husband, when he went to the Annual Conference, to bring that point up and have it voted on. (Her husband died before the conference.)
Yesterday I met a very charming older man at the grocery store who wanted to talk to me about his pets. He was in the last stage of renal failure and though he had plenty of people that wanted his animals he still was concerned that something not so nice might happen to them. He talked about how all 4 slept on his bed (2 dogs and 2 cats) and how each of the dogs had paired up with a cat. Laughter as he talked about how if he got up in the middle of the night the animals always shift their positions so that he has to squeeze in a tighter space to go back to sleep. (His wife had been dead for 25 years). Very sweet.
I don't know if pets have souls. But I think it's arrogant to not think that there is a possibility of it.