The very first verse of the bible tells us that God is the Creator of all things. Life is a wonderful gift. When life is gone, and we see the dead body, we know instinctively that nothing is left but 'dust'. It is not hard to understand why human imagination should have jumped to the conclusion that the life, which has fled, was the 'soul'. But animals too consist of 'dust' and the breath of life, and the same process happens to them when they die.
Most theologians say no.
It would be much too inconvenient to countenance such an idea anyway. Vegetarians might be more inclined to say yes because they are at least consistent in their regard for animals. Christians should be prepared to put aside personal preference or prejudice from whatever background they approach the question and consider God's word on the matter. God's word is clear about this. The 'soul' is the living being, and consists of dust and the breath of life.