It is no secret that the so-called “Western” text, as represented by the distinctive readings of its witnesses,* is not at all confined to the West, but rather may be found across a range of geographical and versional contexts, from the Old Syriac and Coptic versions to various Greek traditions associated with Eastern writers, for example, with Origen or Basil of Caesarea.
G. Zuntz put it well:
“The term ‘Western Text’ is a misnomer — everybody knows that. This characteristic form of the Sacred Text, first discovered in Codex Bezae and the Old Latin, is found in Eastern witnesses as well.” 
B. Aland and K. Aland observe with some hyperbole:
“The origin of the ‘Western’ text lies anywhere but in the direction its name would suggest.” 
But if the “Western” text is “Western” only in its Latin representatives, what are we to think about its origins? Must they lie in the East — whether in Greek or in Syriac?
In fact, if we follow Zuntz, the “Western” text consists of two layers, both with Eastern origins:
- First, there is a base layer containing “remnants of the original text,” which being a Greek tradition, presumably derives from the East.
- Then, there is a secondary layer representing “the product of retranslation, from the Syriac, of an ecclesiastical adaptation, namely, the lectionary-text of the oldest Church of Edessa,” which of course, being Edessene, must have arisen in the East as well. 
According to Zuntz then, the “Western” text form is essentially an “eastern” text form whose most distinctive, secondary elements derive from Syria.
Certainly, there are elements of the “Western” text that seem to demand an Eastern background. At the same time though, it is difficult to overlook the close association of some of the most distinctively “Western” readings with the Old Latin version. Yet since there is no secure evidence of a Latin version before the early third century, to propose a developed “Western” text in the second century all but requires us to place its origins in the East.
But if we postpone the time frame when the elements of the mature “Western” text form came together to the second half of the fourth century, we are now no longer constrained to an Eastern provenance. In fact, with its reputation for Nicene orthodoxy, the West had certain advantages as a promulgator of influence — and perhaps even of textual influence — to the East from the Synod of Alexandria (362) onward, especially from the perspective of the pro-Nicene party.
Thus we find Basil of Caesarea in Cappadocia [!] turning to Rome for assistance in resolving the Antiochian schism:
“to move some of the Italians to undertake a voyage by sea to visit us.” (Epist. 68; NPNF 2.8, 164)
“It has seemed to me to be desirable to send a letter to the bishop of Rome, begging him to examine our condition, and since there are difficulties in the way of representatives being sent from the West by a general synodical decree, to advise him to exercise his own personal authority in the matter by choosing suitable persons to sustain the labours of a journey … to correct the unruly among us here; able to speak with proper reserve and appropriateness, and thoroughly well acquainted with all that has been effected after Ariminum to undo the violent measures adopted there.” (Epist. 69.1; NPNF 2.8, 166)
But should we expect these worthy representatives from the West to have left their gospels in Italy? Would they have stopped at correcting the “unruly” to also correct unruly texts? Would they have spoken with proper “appropriateness” but not appropriately cite from their own orthodox readings?
Of course, we can only speculate as to the potential impact such contact might have had. Yet we may observe one thing with greater confidence, namely, that by the 370’s there was a flow of influence from West to East that was both initiated and received by sympathetic Easterners. In such a context, we can plausibly imagine Latin readings being imported to the region surrounding Antioch and the great library at Caesarea, where it seems witnesses such as Sinaiticus and Bezae might at one point have resided.
References and Notes
* I added the phrase “as represented by the distinctive readings of its witnesses” in response to James Snapp, Jr.’s insightful comment below.
 Günther Zuntz, “The Western Text of the Acts of the Apostles” in Opuscula Selecta; Classica, Hellenistica Christiana (Manchester: University Press, 1972) 189–215 at 189.
 K. Aland and B. Aland, The Text of the New Testament: An Introduction to the Critical Editions and to the Theory and Practice of Modern Textual Criticism (2d ed.; Grand Rapids: Eerdmans, 1987) 67.
 Zuntz, “Western Text,” 189-190.