According to the Bible, do animals have souls? Randy Alcorn claims in his book Heaven (page 374) that animals have souls, though not necessarily like humans. Is there any Biblical support for this?
In the Bible, the word for "spirit" is the same as "breathe." But there is ultimately a difference between the life of humans and the life in animals.

In Genesis 2:7, God breathes into the nostrils of man and gives him life, but all land animals have this same breath as well. "And they went in unto Noah into the ark, two and two of all flesh, wherein is the breath of life." (Genesis 7:15 (KJV))

However, the distinction between the two types of spirits is apparent in the following verses:


  Nevertheless man being in honour abideth not: he is like the beasts
  that perish.This their way is their folly: yet their posterity approve
  their sayings. Selah.Like sheep they are laid in the grave; death
  shall feed on them; and the upright shall have dominion over them in
  the morning; and their beauty shall consume in the grave from their
  dwelling. But God will redeem my soul from the power of the grave: for
  he shall receive me. Selah.
   Psa 49:12-15 (KJV) 
  
  Who knoweth the spirit of man that goeth upward, and the spirit of the beast that goeth downward to the earth?
  Ecc. 3:21 (KJV) 


It is clear that God does care for the animals. In many passages, God shows compassion for the beast or bird, whether by the law, or other means. Though they do not have eternal souls, they are part of God's creation, and they give Him His due worship.