Is this a significant scholarly position?
Significant enough that it is discussed regularly in various scholarly places. For example, there is extensive discussion in a fairly recent work: D. A. Carson and Douglas J. Moo, An Introduction to the New Testament, 2nd ed. (Grand Rapids, MI: Zondervan, 2005), 142-150 (hereafter referred to as C&M).
In that work they discuss the credibility of the earliest witness of authorship, Papias, whose original works are lost, but is quoted by Eusebius in Historia Ecclesiastica 3.39.16 as
ταῦτα μὲν οὖν ἱστόρηται τῷ Παπίᾳ περὶ τοῦ Μάρκου: περὶ δὲ τοῦ Ματθαίου
ταῦτ̓ εἴρηται: ‘Ματθαῖος μὲν οὖν Ἑβραΐδι διαλέκτῳ τὰ λόγια συνετάξατο,
ἡρμήνευσεν δ̓ αὐτὰ ὡς ἦν δυνατὸς ἕκαστος.’
Translated to English in Philip Schaff and Henry Wace, eds., Eusebius: Church History, Life of Constantine the Great, and Oration in Praise of Constantine Vol. 1, A Select Library of the Nicene and Post-Nicene Fathers of the Christian Church, Second Series (New York: Christian Literature Company, 1890), 173:
But concerning Matthew he writes as follows: “So then Matthew wrote
the oracles in the Hebrew language, and every one interpreted them as
he was able.”
As C&M state, however, the passage from Papias "is notoriously difficult to translate," and they give various translation possibilities in parenthesis of this expanded translation of theirs (p.143):
“Matthew συνετάξετο (synetaxeto, ‘composed’? ‘compiled’? ‘arranged [in
an orderly form]’?) τὰ λόγια (ta logia, ‘the sayings’? ‘the gospel’?)
in Ἑβραΐδι διαλέκτῳ (Hebraïdi dialektō, ‘the Hebrew [Aramaic]
language’? ‘Hebrew [Aramaic] style’?), and each ἡρμήνευσεν
(hērmēneusen, ‘interpreted’? ‘translated’ ‘transmitted’?) them as best
C&M further state that (p.143):
There is no doubt that the early church understood this to mean that
Matthew first wrote his gospel in Hebrew or Aramaic (the same Greek
word was used to refer to both cognate languages) and that it was then
translated by others.
Which they later support this view of the church fathers, stating (p.145; note that I have added links to English translations in brackets following links to original text):
for instance, of Irenaeus (Adv. Haer. 3.1.1 [English], quoted in Eusebius, H.E.
5.8.2 [English]), Tertullian (Adv. Marc., 4.2 [this support seems weak to me, as the English translation is not clear that he ascribes Matthews gospel to Hebrew; I do not read Latin]), Origen (quoted by Eusebius, H.E. 6.25.3–6 [English]), Eusebius himself (H.E. 3.24.5–6 [English]), and Jerome (De vir. ill. 3 [English])
Some scholars holding this view are noted by C&M (p.143 n.17):
C. F. Burney, The Poetry of Our Lord (Oxford: Oxford University Press,
1925); C. C. Torrey, Our Translated Gospels (London: Hodder &
Stoughton, n.d.); A. Schlatter, Der Evangelist Matthäus: Seine
Sprache, sein Ziel, seine Selbständigkeit, 6th ed. (Stuttgart: Calwer,
1963); P. Gaechter, Die literarische Kunst im Matthäusevangelium
(Stuttgart: Katholisches Bibelwerk, 1966); J. W. Wenham, “Gospel
Origins,” TrinJ 7 (1978): 112–34.
For one site's listing of quotes from scholarly supporters, see the page on hebrewgospel.com.
What evidence supports it?
The main evidence is from the early church testimony—see links given above and perhaps points as argued on hebrewgospel.com. An issue, however, is whether these are independent witnesses or not. In other words, are all the witnesses that testify to it themselves relying on Papias's testimony, or did they in fact know of a written form in Hebrew themselves. For that matter, did Papias witness such a document himself, or is he also testifying second hand?
The fact is, at present, no ancient extant Hebrew version of Matthew is known to exist confirming the actuality of such a document being the source for the Greek Gospel of Matthew. That does not itself exclude the possibility, but finding such would confirm Papias.
C&M hold that "substantial linguistic evidence is against" the view because of a "mix of text forms" within the gospel and that the gospel does not "read like translation Greek" (143; they also hold to a Markan priority, which if true, complicates things by the similarities between Mark/Matthew).
Have there been attempts to reconstruct the original Hebrew?
Not that I am aware of, but if I run across anything otherwise, I'll add it here.
For those who believe in inspiration, would it then be the Hebrew that is inspired?
This I can answer for myself (since I "believe in inspiration"), though my answer could vary from others also so believing. For me, there is not doubt that the Greek is the inspired text because it is the preserved text. Whatever (if any) background writings were in Hebrew/Aramaic as "source" material, it is the forming of the Greek text that is deemed the point of inspiration (i.e. when the Holy Spirit was involved in moving Matthew to write it), in part because that was the text God preserved through the human copying process.
As an example of another who holds to "inspiration" of the text, yet allows for the possibility of an earlier Hebrew version, John Peter Lange and Philip Schaff, A Commentary on the Holy Scriptures: Matthew, 1879 (Bellingham, WA: Logos Bible Software, 2008) resolve the issue like so (regarding their view of inspiration, see p.11-12):
The Gospel of Matthew, originally written in Hebrew, was translated at
a very early period, and probably by Matthew himself, into our present
Greek Gospel, which has ever since been received as canonical in the
So the "canonical" version is considered the Greek. The quote seems to state they believe there was such an earlier version, however, later they note with respect to this possibility:
an independent examination of our present Greek Gospel by Matthew, and
especially of the independent form of his quotations from the Old
Testament as compared with the Septuagint, leaves the impression of an
original work, whether it was written by Matthew himself, or by some other person clothed with apostolic authority. Papias relates that this Gospel was repeatedly interpreted, and the apostolic Church undoubtedly retained its most trustworthy rendering. This translation was preserved in its purity, and obtained canonical authority; while the Hebrew original was afterwards corrupted and interpolated by the Jewish-Christian sects, and in this heretical form called the Gospel of the Hebrews, which lost or rather never enjoyed canonical authority (p.42)
So the Greek is "an original work" (which implies inspired if they are holding to that) and while the Greek may be a "translation," it is one they state "was preserved in its purity." So that is how they deal with the issue.
I do feel there may be warrant for Matthew having some "source" material in Hebrew/Aramaic. Personally, I think with him being involved as a tax collector (and thus used to keeping records), that he may well have taken "notes" about his time with Jesus contemporaneous to it happening (i.e. a daily/weekly journal, so to speak). This may have been a source of his later inspired writing in Greek. But one can only conjecture about this possibility—the only extant evidence of any text is the Greek we have.