Constantine would have heard read in church, or known that the church accepted, most of the New Testament books we acknowledge today, and perhaps several other books as well. All of them were written well before his time, in the first or second century AD (Constantine first being acclaimed as Emperor in 306 AD).

From the mid-second century, Christians had been making lists of which books they deemed part of the canonical (i.e. authoritative) New Testament. As far as we know, the first such list was made by Marcion around 140 AD, who accepted only ten Pauline epistles and an abridged version of the Gospel of Luke - he deemed the other books heretical; other Christians in turn declared Marcion a heretic. It was perhaps in response to Marcion that other Christian leaders started listing canonical books. The Muratorian Fragment, probably dated c. 170 AD, accepts 22 New Testament books out of today's 27, together with the Apocalypse of Peter (today rejected) and the Wisdom of Solomon (which some denominations today accept as part of the Old Testament, and others reject). It also explicitly rejects several other works such as the Shepherd of Hermas, implying that other churches did consider them canonical.

From then through Constantine's time, as far as we can tell, most Christian churches accepted most of the same New Testament books, with some dispute around some. The church historian Eusebius, an advisor to Constantine, summarizes the state of the canon in his time:

First then must be put the holy quaternion of the Gospels; following them the Acts of the Apostles. After this must be reckoned the epistles of Paul; next in order the extant former epistle of John, and likewise the epistle of Peter, must be maintained. After them is to be placed, if it really seem proper, the Apocalypse of John, concerning which we shall give the different opinions at the proper time. These then belong among the accepted writings.

Among the disputed writings, which are nevertheless recognized by many, are extant the so-called epistle of James and that of Jude, also the second epistle of Peter, and those that are called the second and third of John, whether they belong to the evangelist or to another person of the same name.

[Note these prior two paragraphs, between them, are exactly the same as what modern Christians would call the entire New Testament.]

Among the rejected writings must be reckoned also the Acts of Paul, and the so-called Shepherd, and the Apocalypse of Peter, and in addition to these the extant epistle of Barnabas, and the so-called Teachings of the Apostles; and besides, as I said, the Apocalypse of John, if it seem proper, which some, as I said, reject, but which others class with the accepted books. And among these some have placed also the Gospel according to the Hebrews...

Of Paul's letters, Eusebius says elsewhere:

Paul’s fourteen epistles are well known and undisputed. It is not indeed right to overlook the fact that some have rejected the Epistle to the Hebrews... on the ground that it was not written by Paul...

So, while the canon of the New Testament was not completely settled by Constantine's time, much of it was settled, and it was approaching being settled. By the early fifth century, the Western church would have firmly agreed on the current New Testament canon.