Two transgender New Yorkers say that their birth certificates incorrectly identify their sex, subjecting them to awkward and humiliating public moments. So they are suing the city to change a requirement that transgender people have genital surgery to get their sex changed — corrected, as they say — on their birth certificates.

The lawsuits, filed in State Supreme Court in Manhattan on Tuesday on behalf of Sam Berkley and Joann Marie Prinzivalli, argue that the 40-year-old city health code requiring “convertive surgery” to change the sex on a birth certificate is arcane. Convertive surgery has been interpreted to mean the construction of genitalia, according to the lawsuits, brought by the Transgender Legal Defense and Education Fund. A third transgender resident, Patricia Harrington, has not yet filed papers, but plans to join Mr. Berkley and Ms. Prinzivalli.

“For one thing,” said Noah Lewis, a lawyer for the fund, “it’s expensive, and insurance companies often deny coverage for this.”

Mr. Lewis added: “It’s just a highly individual decision about whether you want to have surgery on your genitals. There are risks with this surgery. People might be more inclined to stick with what they have.”