Mr. Pompeo’s statement did not offer a rationale for why the State Department does not consider Ms. Muthana a citizen. But American officials seem to be hinging their argument against allowing her back in on an exception in the law.

Ms. Muthana’s father was a Yemeni diplomat, and children born in the United States to active diplomats are not bestowed birthright citizenship, since diplomats are under the jurisdiction of their home countries.

That law does not apply in Ms. Muthana’s case, said Charlie Swift, the director of the Constitutional Law Center for Muslims in America, who is representing her family. Ms. Muthana, he said, was born a month after her father was discharged from his position as a United Nations diplomat.

After she joined the Islamic State, Mr. Swift said, Ms. Muthana’s family received a letter indicating that her passport had been revoked. Her father sent the government evidence of his nondiplomatic status at the time of his daughter’s birth, but did not receive a response.

Mr. Swift said Ms. Muthana had in fact been issued two American passports: one when she was a child, and a renewal she applied for herself just before leaving for Syria. In the case of the first, he says that her father provided a letter from the United Nations proving that he had been discharged, to overcome the jurisdictional challenge.

Hassan Shibly, a lawyer with the Council on American-Islamic Relations Florida who is advising the family, provided a birth certificate for Ms. Muthana that showed she was born in Hackensack, N.J., on Oct. 28, 1994.

Mr. Shibly said that her father left the foreign service in June 1994. He later sent a photograph of a document, on the letterhead of the United States Mission to the United Nations and signed by one of its representatives in 2004, that dated the official end of his service to Sept. 1, 1994. Ms. Muthana, Mr. Shibly said, “is trying to turn herself in to federal authorities and face consequences for her actions.”