The canonization of the New Testament writings started with oral tradition focused on the life, ministry and teachings of Jesus of Nazareth. F.F. Bruce writes,
…what he [Jesus] said was treasured and repeated by those who heard him, and by their hearers in turn. To those who confessed him as Lord his words were at least as authoritative as those of Moses and the prophets. They were transmitted as a most important element in the ‘tradition’ of early Christianity, together with the record of his works, his death and resurrection. These were ‘delivered’ by the original witnesses and ‘received’ in turn by others not simply as an outline of historical events but as the church’s confession of faith and as the message the message which it was commissioned to spread abroad.
This sacred oral tradition of Jesus eventually was recorded in written form due to the need to ensure the accuracy of the Jesus tradition. From an evangelistic perspective, written accounts of the life, ministry and teachings of Jesus would have had mass appeal in the Greco-Roman literate society. Also, they would have been helpful in responding to heretical teachings that were infiltrating the early christian movement.
In 1 Timothy 5:18, the apostle Paul connects this early written record of Jesus with the Hebrew scriptures by quoting Luke 10:7 and Deuteronomy 25:4, while using the one phrase, “For the scripture says.” F.F. Bruce writes,
What is striking here is that a saying of Jesus known to us from Luke 10:7 is linked with an Old Testament text under the common rubric: ‘the scripture says’. It has to be considered whether ‘the scripture’ refers strictly to the commandment from Deuteronomy, or also to a written collection of sayings of Jesus which may have served as a source for the Third Evangelist, or even to the gospel of Luke itself. 
While this verse can be debated about its meaning, it may indicate that there was a body of writings focused on the life, ministry and teachings of Jesus that were recognized as being inspired by God and thus included in the canon of scripture.
A similar passage written by the Apostle Peter refers to the writings of the Apostle Paul as Scripture. 2 Peter 3:15-16 states,
Bear in mind that our Lord’s patience means salvation, just as our dear brother Paul also wrote you with the wisdom that God gave him. He writes the same way in all his letters, speaking in them of these matters. His letters contain some things that are hard to understand, which ignorant and unstable people distort, as they do the other Scriptures, to their own destruction.
An interesting aspect of this passage is that it reminds the reader of the integrity of the New Testament authors. The Apostle Peter was rebuked by the Apostle Paul early in the Christian movement, yet the Apostle Peter still was able to refer to the Apostle Paul’s writings as scripture with honesty and trustworthiness. In addition, the Apostle Peter is comparing the Apostle Paul’s writings in the same sense as the Hebrew scriptures referred to earlier in his writing (cf. 2 Pet 3:5, 7-8). With this passage, a logical conclusion can be asserted that in the first century church the Pauline epistles were recognized as inspired by God and thus authoritative and canonical.
Norman Geisler and William Nix summarize the impact that 1 Timothy 5:18 and 2 Peter 3:15-16 have on the canonization of the rest of the New Testament by stating, “Certainly if Paul’s and Luke’s writings were considered Scripture, the the epistles of the apostles of Jesus, and the particularly those of the “inner circle” (Peter and John), which traditionally make up most of the remainder of the New Testament, cannot logically be excluded from the category of inspired Scripture.”
Along with the internal evidence for the canonization of the New Testament writings, is the external evidence of the Early Church Fathers. Throughout the centuries following the death of the apostles, the Early Church Fathers quoted from the writings of the apostles and began to form canons of recognized inspired writings. Also, Church councils and synods were formed in order to address the issue of the canon of scripture. Wegner suggests principles that would have likely assisted in the process of canonization of the New Testament writings. 1) Was the book written by an apostle or at least someone or recognized authority? 2) Did it agree with the canon of Truth? 3) Did it enjoy universal acceptance? 4) Does it have a self-authenticating divine nature?
 F.F. Bruce, The Canon of Scripture. (Downers Grove, IL:, 1988), 118.
 Paul D. Wegner, The Journey from Texts to Translations: The Origin and Development of the Bible. (Grand Rapids, MI.: Baker Academic, 1999), 134-135.
 Bruce, Canon, 120.
 Norman L. Geisler and William E. Nix, A General Introduction to the Bible. Revised and Expanded ed. (Chicago, IL:, Moody Press, 1986), 53.
 Ibid., 53.
 Wegner, The Journey, 148.