Genesis 2:5-7 states: "Now no shrub had yet appeared on the earth and no plant had yet sprung up, for the Lord God had not sent rain on the earth and there was no one to work the ground, but streams came up from the earth and watered the whole surface of the ground. Then the Lord God formed a man from the dust of the ground and breathed into his nostrils the breath of life, and the man became a living being."
God breathed into his nostrils the breath of life. This is oftentimes interpreted as being the soul. However, notice that the breath of life made man live; it is therefore reasonable to assume that anything that lives has the breath of life. As such, animals have souls because of the very fact that they live.
According to the Quran, a believer has to complete the Five Pillars of Islam. Http://en.Wikipedia.Org/wiki/Five_pillars_of_islam#The_Five_Pillars_of_Islam As one can see, animals are incapable of doing any of those things.
According to the Bible, a person has to accept Jesus as his savior. Animals seem incapable of doing this as well. I don't see how an animal would even know who Jesus is...
Animals do not have souls. Actually, humans don't even have souls. The idea of the soul was added to Christianity from Greek philosophy. The writers of the Bible didn't believe in souls, and the misconception eventually found its way into Islam.