The fact that all people have a sinful nature IS a Bible teaching. We have listed some of those verse below. However, that is NOT the same thing as the Catholic doctrine of "Original Sin". The following excerpt is from The Berean Call, Volume XXIX No. 6, June 2014 and is most helpful in identifying the issue:
"Many Christians make the mistake of thinking that "original sin" in Roman Catholic theology equates to and is just a different way of describing the sin nature. This is an utter misconception with eternal consequences. According to the RCC, Adam and Eve were created possessing "sanctifying grace" - which was "something" that was lost when they sinned against God. Likewise their descendants inherit original sin - but they don't believe it is something that people are born with. Rather, there is something that people are born without - namely, sanctifying grace. Therefore, the solution to the original sin problem is to transfer sanctifying grace back into the soul - which takes place through the vehicle of the sacraments, the first of which is the sacrament of baptism. This is why the baptism of infants makes perfect sense within Catholic theology."
"Interestingly, this meshes perfectly with Augustine's theology that we aren't inherently evil; we simply lack sanctifying grace - a problem that is easily solved through receiving the sacraments. Unfortunately for the Catholic, over time, sanctifying grace diminishes - and can be partially lost (through venial sins) or completely lost (through mortal sin). And it is this matter of venial and mortal sin that reveals one of the most tragic aspects of Catholic theology."
"We understand that the Bible teaches that although sins have varying degrees of consequences, any and every sin makes us sinners before a holy God and unfit for heaven. Yet, the Bible also teaches that we are cleansed and forgiven of every sin the moment we trust in Christ for salvation."
"In contrast, Catholic theology teaches that although Christ's death made salvation possible, it is up to each individual to earn the right to enter heaven. Those who die with only venial sin on their soul must spend time in the fires of purgatory, literally paying the penalty for their own sins, after which they will eventually make it to heaven. However, if someone dies with a mortal sin on their soul, meaning they have lost all sanctifying grace, they will spend eternity in hell."
"But it gets worse - no one (not even the pope) can tell you whether or not a sin is a venial or mortal sin. There is no list. It depends not only on the seriousness of the sin but upon a number of other factors, including the circumstances and the attitude at the time the sin was committed. And in fact, no one can know until after they die which were which. This is why Catholics are slaves to the sacramental system and why they can have no assurance (which the Bible provides) of eternal salvation by grace through faith in Christ alone."
Bible verses regarding man's sinful nature:
Jeremiah 17:9 "The heart is deceitful above all things, and desperately wicked: who can know it?"
Romans 5:12 "Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned."
Romans 7:23 "But I see another law in my members, warring against the law of my mind, and bringing me into captivity to the law of sin which is in my members."
Galatians 3:22 "But the Scripture has concluded all under sin, that the promise by faith of Jesus Christ might be given to them that believe."
Links regarding widely held view regarding sinful nature and total depravity:
http://members.cox.net/hughdelong2/studynotes/bibl157.htm (Part 1 of 3)
http://members.cox.net/hughdelong2/studynotes/bibl158.htm (Part 2 of 3)
http://members.cox.net/hughdelong2/studynotes/bibl159.htm (Part 3 of 3)
Links regarding an alternative view which allows for "total depravity" without "inheriting" the consequences of Adam's sin: