Calvinist interpretation of John 6.37-45 is contrary to the entire tenor of Scripture. Let's compare it with John 6.65: "And he said, Therefore said I unto you, that no man can come unto me, except it were given unto him of my Father."
"All that the Father giveth me shall come to me; and him that cometh to me I will in no wise cast out. For I came down from heaven, not to do mine own will, but the will of him that sent me. And this is the Father's will which hath sent me, that of all which he hath given me I should lose nothing, but should raise it up again at the last day. And this is the will of him that sent me, that every one which seeth the Son, and believeth on him, may have everlasting life: and I will raise him up at the last day. The Jews then murmured at him, because he said, I am the bread which came down from heaven. And they said, Is not this Jesus, the son of Joseph, whose father and mother we know? how is it then that he saith, I came down from heaven? Jesus therefore answered and said unto them, Murmur not among yourselves. No man can come to me, except the Father which hath sent me draw him: and I will raise him up at the last day. It is written in the prophets, And they shall be all taught of God. Every man therefore that hath heard, and hath learned of the Father, cometh unto me." (John 6.37-45)
In 6.65 John says "no man can come unto me, except it were given [Greek, didomi] unto him of my Father." Note this is not a giving of the sinner to the Son, but a giving to the sinner (given him), making it possible for him to come to Christ. The Father gives the opportunity to come, i.e. the Father drawing. We have an abundance of scripture indicating that this opportunity is given to the whole world through the gospel.

The very same Greek word (didomi) is used for "given" multiple times in the New Testament in a way that allows a distinctly non-Calvinist interpretation of Christ's words here, and which is also consistent with the overall biblical emphasis upon God's love and mercy. Paul uses didomi when he says that God "giveth to all life, and breath, and all things" (Acts 17.25). There are many places where didomi is used to indicate something given by God, and which men can either receive or reject, obey or disobey, and which involves their cooperation:
  • The law was given by Moses...(John 1.17). Nobody was forced to obey, although there is serious consequences for disobedience.
  • I would have given thee living water (John 4.10). The water would not be forced upon her against her will. She would have to want it and willing drink it.
  • I have given them my word...(John 17.14). The disciples had to willing receive the Word and obedience thereto was by their choice-it wasn't forced upon them.
  • The cup which my Father hath given me, shall I not drink it? (John 18.11). Jesus pleaded with the Father that if salvation could come to mankind any other way to spare Him this cup. However, He drank it out of obedience to the Father and love for us.
  • Through the grace given unto me...(Rom. 12.3, 15.15; Gal. 2.9). Paul uses this expression with the same Greek word a number of times. This is not Calvinism's mythical Irresistible Grace. God's grace was not imposed upon him so that he could not disobey or fail to fulfill all God's will, or did not need to cooperate in the fulfillment thereof.
The Father draws the lost to Christ by giving (didomi) to them the opportunity to believe. The giving of those who believe to the Son is of another nature. And those who are drawn by the Father must, in response to the Father's drawing, "see" Him with the eyes of faith and believe on Him to be saved. The giving of the redeemed by the Father to the Son is something else-a special blessing for those who believe.