TT #56 · Parshas Tazria · Aliyah 1 · Passuk 8
English: If, however, her means do not suffice for a sheep, she shall take two turtledoves or two pigeons, one for a burnt offering and the other for a purgation offering. The priest shall make expiation on her behalf, and she shall be pure.
Chabad.org — Chumash, English translation + Rashi for Vayikra 12:8
Torah Temimah cites Yoma 41a for the scriptural basis of the bird-offering designation rule.
E-Daf — Yoma 41a · Shas.org · Sefaria — source text and translation
English: Rav Shimi bar Ashi explains Rav Chisda’s rule from the verse: “she shall take two turtledoves” and “the priest shall make”—designation occurs at the owner’s taking or at the priest’s service.
Legal designation
The two phrases in the verse supply the legal basis for the two possible moments of designation: the owner’s taking or the priest’s service.