R. Joseph B. Soloveitchik, Kinos Mesoras HaRav, p. 370:
It is noteworthy that the Western Wall is not referred to at all in the Babylonian Talmud or the Jerusalem Talmud and is hardly mentioned in the Rishonim. For example, Maimonides’ letter describing his arrival in Jerusalem does not mention anything about the Western Wall. There is a reference to the Western Wall in the Midrash on the verse from Song of Songs, “Behold, he stands ahar kotlenu, behind our wall, He looks in through the windows, He peers through the lattice” (2:9). The Midrash says (Shemot Rabba 2:2) that this refers to the Western Wall and that the Shekhina is behind the Western Wall, in its shadow. I have always been somewhat skeptical of the authenticity of the midrashic references to the Western Wall, and I suspect they may be of a later period, because the classical Talmudic sources make no mention of the Western Wall. In my view, this kina of Rabbi Elazar HaKalir is one of the earliest documents in which the Western Wall is mentioned.