Since we were created by God and in His image, how is it possible that we have a sinful nature?
The Bible doesn't use the phrase "sinful nature." It is the New International Version that substitutes "sinful nature" for the word "flesh" in many places. In doing so, it introduces a concept not taught in the Bible. It implies that a person sins because of how he is made. But the Bible states that sin is breaking the laws of God. "Whoever commits sin also commits lawlessness, and sin is lawlessness" (I John 3:4). One does not sin merely by existing. If such were true, then Jesus would be sinful since he was made in the likeness of man. "Inasmuch then as the children have partaken of flesh and blood, He Himself likewise shared in the same, that through death He might destroy him who had the power of death, that is, the devil, ... Therefore, in all things He had to be made like His brethren, that He might be a merciful and faithful High Priest in things pertaining to God, to make propitiation for the sins of the people" (Hebrews 2:14, 17). "For we do not have a High Priest who cannot sympathize with our weaknesses, but was in all points tempted as we are, yet without sin" (Hebrews 4:15). The conclusion is that sin does not originate in our physical bodies.
The Bible also states: "Behold, I have found only this, that God made men upright, but they have sought out many devices." (Ecclesiastes 7:29). We start out righteous, but then we sin. Paul mentioned this in his own life: "I was once alive apart from the Law; but when the commandment came, sin became alive and I died; and this commandment, which was to result in life, proved to result in death for me; for sin, taking an opportunity through the commandment, deceived me and through it killed me" (Romans 7:9-11). When would Paul or any other person be apart from the Law? The answer would be only in childhood when the concept of Law and the accountability to Law is not yet formed. This is why God did not hold the children in Israel responsible for their parents' sin. "Moreover, your little ones who you said would become a prey, and your sons, who this day have no knowledge of good or evil, shall enter there, and I will give it to them and they shall possess it" (Deuteronomy 1:39). They had no knowledge of good or evil -- the exact same phrase for the tree of the knowledge of good and evil in the Hebrew. Therefore, children are born sinless but fall prey to sin when they reach an age to understand good and evil.
Since we were not made sinful and we are not sinful as children, it means we are not by nature sinful.
People sin because they choose to sin. "Therefore, just as through one man sin entered the world, and death through sin, and thus death spread to all men, because all sinned" (Romans 5:12). Adam started it, and we all repeat it in our own way. "Nevertheless death reigned from Adam to Moses, even over those who had not sinned according to the likeness of the transgression of Adam, who is a type of Him who was to come" (Romans 5:14).
The source for the temptation to sin is based on our flesh. "But each one is tempted when he is drawn away by his own desires and enticed. Then, when desire has conceived, it gives birth to sin; and sin, when it is full-grown, brings forth death" (James 1:14-15). That is why Jesus took of flesh -- to prove that no one had to sin. He faced the same temptations we all do, yet without sin. The difference between Jesus and the rest of mankind is the Jesus did not choose to sin and the rest of us give in. "For all have sinned and fall short of the glory of God" (Romans 3:23). It is because of Jesus' sacrifice on our behalf that we don't have to remain in our sins.