Ordinarily “blood of the covenant” דם ברית
was understood to refer to the blood of circumcision.
Jerusalem Talmud Yebamot 8.9A.5: Whoever has pulled forth the foreskin
(and thus made his circumcision unrecognizable), whoever is born circumcised
and whoever had himself circumcised before his conversion to Judaism, one must
make the blood of the covenant drip from him (by cutting into the site of
circumcision). R. Simeon b. Eleazar (ca. 190) taught, “The schools of Shammai
and Hillel were not of different opinions about the fact that one must make the
blood of the covenant drip from one who is born circumcised, because his
foreskin is pressed down. They were of different opinions concerning the
proselyte who converts to Judaism already circumcised; for the school of
Shammai said that one must make the blood of the covenant drip from him,
whereas the school of Hillel said that one did not have to do this.”—Parallel
passages include t. Šabb. 15.9 (133); y. Šabb. 19.17A.39; b. Šabb. 135A; b.
Yebam. 71A; Gen. Rab. 46 (29B). (Hermann L. Strack and Paul
Billerbeck, A Commentary on the New Testament from the Talmud and Midrash,
ed. Jacob N. Cerone, 4 vols. [trans. Andrew Bowden and Joseph Longarino;
Bellingham, Wash.: Lexham Press, 2022], 1:1140)
