The theological meaning of "draw" is in the figurative sense. It means "to attract by the expression of loyal love or covenant loyalty." It is not synonymous with salvation.

Before Jesus, God drew men by demonstrating His love and lovingkindness.

The reality of the Abrahamic Covenant was perpetuated by God's faithful commitment to it through blessings (i.e. Jews are the only small ethnic group that can trace its roots back to its beginning, etc.) and miracles (i.e. the Exodus, etc.). This is further evidenced by the supreme elaboration of the unilateral and unconditional Abrahamic Covenant with the New Covenant and with the foreknowledge of Israel's failure to uphold their commitment to the Law of Moses (Jer 31:31-33).

Coming to faith in God was available to all, because Gentile Believers could become Jews living under the Law of Moses (Num 15:13-16; Ruth 1:15-17). Jesus corroborates the fact that a "Jew" was not a tribal group within the ethnic race of Hebrews, but instead a genuine Believer of God (John 8:31-47; Rev 2:9; 3:9).

When Jesus arrived on earth, He said, "no one can come to Me unless the Father who sent Me draws him; and I will raise him up on the last day (John 6:44).

This statement was directed exclusively towards disbelieving Jews and in particular Jewish religious authorities; it was not a statement to all non-Believers. Those who knew and obeyed God's word (covenant keeping Believers under the Law of Moses) listened to angels or the Holy Spirit who showed them who the Messiah was (i.e. the one of virgin birth divinely named Jesus ["God is salvation"] in fulfillment of the prophetic name Immanuel ["God with us"] Matt 1:21-23; Is 7:10-14).

Only genuine believing Jews living under the Law recognized that Jesus was the Messiah, because He fulfilled the Old Testament prophecies that spoke of Him (John 5:37-47).

Only genuine believing Jews living under the Law recognized that Jesus was the Messiah, because He did miracles that they knew the only living God could do (John 5:36).

Only genuine believing Jews living under the Law would believe the testimony of John the Baptist who personally saw and listened to God indicating who His Son was (John 5:33-35).

Disbelieving Jews could not recognize any of the works of God, because the means of God's drawing, faithful knowledge of God's word, identified who the Messiah was, which was the stumbling block for the Jews (John 6:45).

In contrast, Gentiles who came to faith in Jesus as the Messiah, did so by simply believing in His claim of being the Messiah (John 4:7-54). Jews who initially did not believe that Jesus was the Messiah, like the Pharisee Nicodemus (John 3:1-2), also came to faith by believing in Jesus' claim of Messiah (John 3:9-21).

For deeper study:

What does it mean "God draws"?

The Bread of Life Chiasm in John 6:35-51

Prepositions… for example "after"

Who are the people of faith in the Bible?

Strangers, aliens, foreigners, and gentiles… oh my!