I see a few different uses of the term prevenient grace in questions here, so I will define what I meant first: a belief held by traditional Baptists and other non-reformed evangelicals, that though we are all totally depraved and unwilling to come to Christ on our own free will, God gives us a enabling grace that allows us to come to regeneration and faith. But this grace can be resisted and ignored.
What is the Biblical basis for this type of grace?
The doctrine of Prevenient Grace is seen by those who adhere to it as a natural outcome of sound Biblical exegesis.
In one respect it's akin to the doctrine of the Trinity in that it resolves apparent discrepancies. With the Trinity, we have clear teachings in Scripture that there is only one God, other clear teachings that Jesus is God, and others supporting that the Holy Spirit is fully God.
To answer this fully is going to take a bit of time, so I'm going to break this down into three sections: Principles that are used when looking at these Scriptures, the supporting Scriptures, and the application of the principles that lead to the conclusion of Prevenient Grace.
Principles
To understand how Prevenient Grace is drawn out of the texts, you need to start by bearing in mind that to these groups, there can be no question that the Bible is inspired, inerrant, and infallible. In other words, it was written by God Himself, using the authors as instruments, and therefore not only does not contain errors, it cannot contain errors. This means that it cannot have contradictions. Something cannot be logically true and false at the same time. Two polar opposite views that contradict each other cannot be true.
Next, you need to understand that if the Bible contains apparent contradictions, then either the bible is not inerrant, or there's a way to resolve the apparent discrepancy.
Supporting Scriptures
The supporting Scriptures here are, interestingly enough, the ones that show the apparent discrepancy:
The verses that support the Calvinistic view that man plays no part in his salvation, or choice are already documented in many questions on this site:
If you take the time to go through the above questions and the answers, the support for the idea that man has no choice in his salvation, and that God chooses who will be saved is overwhelming.
However, on the other side, there is also overwhelming Scriptural support for the idea that we must choose to accept God's free gift of salvation - in other words, man does play a part. (Emphasis mine in the following verses)
I'm going to stop there. A comprehensive list is far too long for an answer here, and that's enough to make the point: There is Scriptural support for the idea that man does have to choose to follow Christ, and that the choice does exist. It is real.
Taking one more tiny (well, maybe not so tiny since it's the central point of all Christianity) piece of evidence into account... Since man cannot save himself, God provided the payment for our sins.
In other words, when man is incapable. God provides a way...
Application of the Principles to the Scriptures to draw out the Biblical Concept
Thus far, we've established the following two opposing facts:
Therein lies the conflict. Do we have a choice or don't we? Are we capable of choosing or aren't we?
There is one other conflict that ties in here, for which I haven't provided the Scriptural arguments, but which bears on this topic: Does God want only certain people to be saved, or does God desire that all would come to Him?.
Assuming that God's Word is true and that he desires that all come unto repentance...
Scriptural facts:
Can be rephrased as...
God desires that all be saved, and he teaches that it's our choice to do so. He doesn't force us. But because we are totally depraved, we cannot choose to follow God. we are utterly incapable. Since we cannot choose to follow Him on our own, but He expects us to choose, He provides a way that allows us to choose to follow God. He calls us first and gives us the previnent grace to choose to obey His perfect will.
...with no conflicts and no leaps of logic.
The idea that God draws and offers salvation to all men, not only a chosen elect, comes from passages such as John 3:16-21 (emphasis mine (note that this passage can just as easily support a Calvinist view)) -
“For God so loved the world, that he gave his only Son, that whoever believes in him should not perish but have eternal life. For God did not send his Son into the world to condemn the world, but in order that the world might be saved through him. Whoever believes in him is not condemned, but whoever does not believe is condemned already, because he has not believed in the name of the only Son of God. And this is the judgment: the light has come into the world, and people loved the darkness rather than the light because their works were evil. For everyone who does wicked things hates the light and does not come to the light, lest his works should be exposed. But whoever does what is true comes to the light, so that it may be clearly seen that his works have been carried out in God.”
In John 12, Jesus says "I, when I am lifted up from the earth, will draw all people to myself."
We see here our total depravity ("people loved the darkness rather than the light"), God drawing all men (prevenient grace), and some choosing to reject Him. This is only a primer of sorts; if you take these verses alone (ignoring or rejecting other passages) you end up denying the sovereignty of God, as it looks like God doesn't always get His way. Arminianism does not sacrifice the sovereignty of God for the free will of man.
I had to research this more before answering. I had never heard of the term before. Really what you are looking for then is "Do you believe that God draws us to him with Holy spirit?".
I am no expert on the matter, but with my knowledge of the scriptures my answer is an emphatic yes! God is known to draw those who have qualities he desires. He opens their heart to the knowledge of himself. A good example of this is in Acts 16:13-15:
"On the Sabbath day we went outside the gate beside a river, where we thought there was a place of prayer, and we sat down and began speaking to the women who had assembled. 14 And a woman named Lyd′i·a, a seller of purple from the city of Thy·a·ti′ra and a worshiper of God, was listening, and Jehovah opened her heart wide to pay attention to the things Paul was saying. 15 Now when she and her household got baptized, she urged us: “If you have considered me to be faithful to Jehovah, come and stay at my house.” And she just made us come."
Some more scriptural Evidence is found at John 6:44:
"No man can come to me unless the Father, who sent me, draws him, and I will resurrect him on the last day."
In fact without God giving us the blessing of his holy spirit, we could not understand him or the blessings he gives us. 1 Corinthians 2:11-13:
"For who among men knows the things of a man except the man’s spirit within him? So, too, no one has come to know the things of God except the spirit of God. 12 Now we received, not the spirit of the world, but the spirit that is from God, so that we might know the things that have been kindly given us by God. 13 These things we also speak, not with words taught by human wisdom, but with those taught by the spirit, as we explain spiritual matters with spiritual words."
God knows that just as we did not come into the truth on our own, we will not stay in the truth on our own. 1 Peter 1:4-5 and Jude 1:24:
4 "to an incorruptible and undefiled and unfading inheritance. It is reserved in the heavens for you, 5 who are being safeguarded by God’s power through faith for a salvation ready to be revealed in the last period of time."
24 "Now to the one who is able to guard you from stumbling and to make you stand unblemished in the sight of his glory with great joy"
Some other scriptures that support this idea are 2 Thessalonians 2:13
"However, we are obligated always to thank God for you, brothers loved by Jehovah, because from the beginning God selected you for salvation by sanctifying you with his spirit and by your faith in the truth."
"For the Son of man came to seek and to save what was lost.”
"For Jehovah searches through all hearts, and he discerns every inclination of the thoughts. If you search for him, he will let himself be found by you, but if you leave him, he will reject you forever."
There are other scriptures, but this post is already monstrous, so I will let this be enough.
God draws us to him, but he doesn't force us to follow and learn about him. He draws us and then gives us to choice or opportunity. Deuteronomy 30:19-20:
"I take the heavens and the earth as witnesses against you today that I have put life and death before you, the blessing and the curse; and you must choose life so that you may live, you and your descendants, 20 by loving Jehovah your God, by listening to his voice, and by sticking to him, for he is your life and by him you will endure a long time in the land that Jehovah swore to give to your forefathers, Abraham, Isaac, and Jacob.”
"This means everlasting life, their coming to know you, the only true God, and the one whom you sent, Jesus Christ."
Some older puritan writers used to talk about a sinner being awakened (not saved, just awakened). This would involve conviction of sin. It is pertinent to the discussion on pre-venient grace (as in the question) as this was the term used by the Puritans for it.
e.g. from John Bunyan's Pilgrim's Progress (emphasis mine): 'This miry Slough is such a place as cannot be mended; it is the descent whither the scum and filth that attends conviction for sin doth continually run, and therefore is it called the Slough of Despond: for still as the sinner is awakened about his lost condition, there ariseth in his soul many fears, and doubts, and discouraging apprehensions, which all of them get together, and settle in this place; and this is the reason of the badness of this ground.' http://www.ccel.org/ccel/bunyan/pilgrim.html
And following is a short excerpt from a Puritan poem (Taken from "The Valley of Vision", published by Banner of Truth) indicating the idea of awakening:
The Awakened Sinner
O my forgetful soul,
Awake from the wandering dream;
Turn from chasing vanities
Look inward, forward, upward,
View thyself,
Reflect upon thyself,
Who and what thou are, why here,
What thou must soon be.
Thou art a creature of God,
Formed and furnished by him,
Lodged in a body like a shepherd in his tent;
Dost thou not desire to know God’s ways?
.....and it goes on for several more stanzas.
But the idea is there about an awakening before actual salvation, or as you term it, pre-venient grace. Some say Isaiah 52 teaches this doctrine.
