New York City health officials proposed on Tuesday that Orthodox Jewish parents be required to sign a consent waiver before they can take part in a circumcision ritual that is believed to have led to the deaths of at least two babies in the city over the past decade.

The proposal, introduced at a Board of Health meeting, represents an escalation of the city’s efforts to curtail the ancient Jewish procedure of metzitzah b’peh, in which an adult male, usually the circumciser, places his mouth directly on the wound created by the removal of the infant’s foreskin to suck away the blood.

Last week, the federal Centers for Disease Control and Prevention published a report based on city information that said that from 2000 to 2011, 11 newborn babies in New York contracted the herpes simplex virus after the ritual. Ten of those babies were hospitalized; two suffered brain damage, and two died.

Based on those findings, the city’s health department issued a statement last week strongly urging that direct oral-genital suction not be performed during circumcision. It also announced that a number of hospitals had agreed to distribute a pamphlet to parents considering at-home circumcision, warning them of the risks.