There is nothing in Genesis 1, 2 and 3 to indicate man was created with an immortal soul. The word soul means a breathing creature...
Gen 2:7 And the LORD God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul.
soul:
H5315
נפשׁ
nephesh
neh'-fesh
From H5314; properly a breathing creature, that is, animal or (abstractly) vitality; used very widely in a literal, accommodated or figurative sense (bodily or mental): - any, appetite, beast, body, breath, creature, X dead (-ly), desire, X [dis-] contented, X fish, ghost, + greedy, he, heart (-y), (hath, X jeopardy of) life (X in jeopardy), lust, man, me, mind, mortality, one, own, person, pleasure, (her-, him-, my-, thy-) self, them (your) -selves, + slay, soul, + tablet, they, thing, (X she) will, X would have it.
Here are a few other "souls"...
Gen 1:20 And God said, Let the waters bring forth abundantly the moving creature that hath life, and fowl that may fly above the earth in the open firmament of heaven.
Here the word life is derived from H5315, nephesh, a soul. so moving creatures in the water are living souls. Anything here to indicate that crabs, clams and sea urchins are immortal?
Gen 1:21 And God created great whales, and every living creature that moveth, which the waters brought forth abundantly, after their kind, and every winged fowl after his kind: and God saw that it was good.
Same word here for creature.
Gen 1:24 And God said, Let the earth bring forth the living creature after his kind, cattle, and creeping thing, and beast of the earth after his kind: and it was so.
Again, creature here is rendered from H5315, nephesh. This would include, as my wife would say, "rats, bats and old cats." They have immortal souls do they?
Gen 1:30 And to every beast of the earth, and to every fowl of the air, and to every thing that creepeth upon the earth, wherein there is life, I have given every green herb for meat: and it was so.
Here the word life is again, nephesh.
We've already seen that Adam was a breathing creature...
Gen 2:7 And the LORD God formed man of the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul.
H5315, nephesh.
Gen 2:19 And out of the ground the LORD God formed every beast of the field, and every fowl of the air; and brought them unto Adam to see what he would call them: and whatsoever Adam called every living creature, that was the name thereof.
Here Adam is naming the living souls, lions, muskrats, vultures and skunks are included.
Now forget what tradition and some man with his collar on backwards has told you, what does God say can happen to a soul?
Gen 2:16 And the LORD God commanded the man, saying, Of every tree of the garden thou mayest freely eat:
Gen 2:17 But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die.
die:
H4191
מוּת
mûth
mooth
A primitive root; to die (literally or figuratively); causatively to kill: - X at all, X crying, (be) dead (body, man, one), (put to, worthy of) death, destroy (-er), (cause to, be like to, must) die, kill, necro [-mancer], X must needs, slay, X surely, X very suddenly, X in [no] wise.
God says that souls can die. He doesn't say that if you sin you will burn forever in hell, He says you will die.
Gen 3:4 And the serpent said unto the woman, Ye shall not surely die:
Here is the Devil saying that you won't really die, you have an immortal soul that lives on. God says you die, Satan says you have an immortal soul. Believe what you will, but I gotta go with God on this one.