Does the Bible say that animals are souls?

"Soul", in the original-language terms ("Ne´phesh" in Hebrew and "psy·khe´" in Greek) means a person, an animal, or the life that a person or an animal enjoys.

The common misconception is that a “soul” is a part of man that separates from the body at death and goes on living. But the Bible shows that Man is a soul:

Quoting Gen. 2:7, 1 Cor. 15:45 says: "It is even so written: `The first man Adam became a living soul.'" (This shows that the Christian Greek Scriptures agree with the Hebrew Scriptures as to what the soul is.)
Both ne´phesh and psy·khe´ are also used to mean life—not merely as an abstract force or principle - but life as a creature, human or animal.
 
The following are a few scriptures showing that animals are souls:

Gen. 1:20, 21, 24, 25: "God went on to say: `Let the waters swarm forth a swarm of living souls . . . ' And God proceeded to create the great sea monsters and every living soul that moves about, which the waters swarmed forth according to their kinds, and every winged flying creature according to its kind. . . . And God went on to say: `Let the earth put forth living souls according to their kinds . . . ' And God proceeded to make the wild beast of the earth according to its kind and the domestic animal according to its kind and every moving animal of the ground according to its kind." (In Hebrew, the word for "soul" here is ne´phesh. Some translations use the rendering "creature[s].")

Lev. 24:17, 18: "In case a man strikes any soul [Hebrew, ne´phesh] of mankind fatally, he should be put to death without fail. And the fatal striker of the soul [Hebrew, ne´phesh] of a domestic animal should make compensation for it, soul for soul." (See that the same Hebrew word for soul is applied to both mankind and animals.)

Rev. 16:3: "It became blood as of a dead man, and every living soul died, yes, the things in the sea."

For more, see
Do You Have an Immortal Soul?

“Soul” and “Spirit”—What Do These Terms Really Mean?