"Just as Jannes and Jambres opposed Moses, so also these teachers oppose the truth. They are men of depraved minds, who, as far as the faith is concerned, are rejected."
Who were they?
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They seem to be of the magicians that worked against Moses in Egypt:
They also appear in Pliny the Elder's Natural History, together with Moses, as famous magicians of antiquity; Pliny's citation is also referred to in Apuleius. Numerius, a Pythagorean philosopher, calls them sacred Egyptian scribes.
This has been taken from Wikipedia, and I don't know how much weight should be given it. However, they seems to be evidence backing it up:
The Apocryphon of Jannes and Jambres the magicians is a manuscript among the Chester Beatty Papyri No XVI, which has been edited and translated by Albert Pietersma.
The Jews have long since known of them. Taken from a Jewish Encyclopedia:
Names of two legendary wizards of Pharaoh "who withstood Moses" (II Tim. iii. 8) by imitating "with their enchantments" the works of Moses and Aaron, though they were defeated (Ex. vii. 11, viii. 7). According to rabbinical tradition they were the two chiefs of the magicians at the court of Pharaoh who foretold the birth of Moses, "the destroyer of the land of Egypt," thereby causing the cruel edicts of Pharaoh (Soṭah 11a; Sanh. 106a). They said to Moses when he performed his miracles with the water and the rod: "Dost thou wish to introduce magic into Egypt, the native land of the magic art?" (Men. 85a)
JANNES AND JAMBRES. Paul speaks of certain false and morally dangerous teachers as resisting the truth as ‘Jannes and Jambres’ resisted Moses (2 Tim. 3:6-8). These names do not occur in OT, but extra-biblical allusions show that the Egyp. magicians of Ex. 7-8 are intended. Like them, the teachers played on superstitious susceptibilities with a plausibly presented parody of the truth.
The names, of unknown age, occur in various forms. The so-called ‘Zadokite Work’, now known to belong with the Qumran literature, has Belial raising up ‘Yoḥaneh and his brother’ against Moses and Aaron (7.19 in R. H. Charles, Pseudepigrapha, 1913; 5.19 in C. Rabin, The Zadokite Documents2, 1958, p. 21); the Babylonian Talmud ’Yoḥanē and Mamre’ (Menaḥoth 85a; cf. the spelling ‘Mambres’ in most Lat. and some Gk. mss of 2 Tim. 3:8). Jewish legend made much of them, even attributing their paternity to Balaam. Pagan sources refer, not always perspicuously, to one or both (cf. Pliny, NH 30. 1. 11; Apuleius, Apology 90; Numenius of Apamea in Eusebius, Praep. Ev. 9. 8. 1), reflecting the story’s celebrity. Origen knew a book on the subject (Comm. in Mt. 23:37; 27:9), and the Gelasian Decree a Penitence of Jannes and Jambres, of which M. R. James identified a fragment in a Saxon ms (JTS 2, 1901, pp. 572ff.). It is improbable, however, that Paul is alluding to the book: he would employ the names simply as being then in common use, with Ex. 7-8 alone in mind.
Wood, D. R. W. ; Marshall, I. Howard: New Bible Dictionary. 3rd ed. Leicester, England; Downers Grove, Ill. : InterVarsity Press, 1996, S. 543
(Jan′nes). A resister of Moses with whom Paul compares apostates who resist the truth. (2Ti 3:8, 9) Jannes and Jambres, whose ‘madness became plain to all,’ are not identified in the Hebrew Scriptures, but it is generally agreed that they were two of the leading men in Pharaoh’s court, perhaps the magic-practicing priests who resisted Moses and Aaron on their numerous appearances there. (Ex 7:11, 12, 22; 8:17-19; 9:11) The amount of tradition that agrees with this greatly outweighs what little there is to the contrary. Non-Christian sources, such as Numenius, Pliny the Elder, Lucius Apuleius, a Qumran writing, the Targum of Jonathan, and several apocryphal writings all mention one or both of these men.
Insight on the Scriptures Vol. 1 pp. 1254-1255 Jannes
JANNES AND JAMBRES Two of Pharaoh’s magicians, who opposed Moses and tried to show that they were as effective as he at working miracles (Ex 7–9). Jewish legend regarded Jannes and Jambres (somewhat improbably) as sons of Balaam, the Midianite prophet of Numbers 22–24. Curiously, the Exodus chapters do not identify them by name. The only biblical reference to them appears in the NT. The apostle Paul saw similarity between Jannes and Jambres and the false teachers of debased intellect who were enemies of the truth in his day (2 Tm 3:6–8).
Much speculation has arisen about the two names. They are apparently Semitic, but their precise derivation is unclear. They are referred to in the Qumran documents and in late Jewish, pagan, and early Christian literature. Variations include “Yohanneh and his brother” (Qumran), “Yohane and Mamre” (Babylonian Talmud), and “Mambres” (the translation in most Latin and some Greek manuscripts of 2 Tm 3:8). The names appear also in the writings of Pliny (first century ad) and of Apuleius and Numenius (both second century), though both names are not always cited.
Origen, an Alexandrian church father, twice referred to an apocryphal work entitled The Book of Jannes and Jambres, suggesting that it was the source of Paul’s words in 2 Timothy. A Latin church document called the Gelasian Decree (fifth or sixth century?) mentions Penitence of Jannes and Jambres, possibly the work mentioned by Origen.
Elwell, Walter A. ; Comfort, Philip Wesley: Tyndale Bible Dictionary. Wheaton, Ill. : Tyndale House Publishers, 2001 (Tyndale Reference Library), S. 669