The New York Times reports he is known as Abdullah al-Shami, an Arabic name meaning Abdullah the Syrian. But his nom de guerre masks a reality: He was born in the United States, and the United States is now deciding whether to kill him. Mr. Shami, a militant who American officials say is living in the barren mountains of northwestern Pakistan, is at the center of a debate inside the government over whether President Obama should once again take the extraordinary step of authorizing the killing of an American citizen overseas. It is a debate that encapsulates some of the thorniest questions raised by the targeted killing program that Mr. Obama has embraced as president: under what circumstances the government may kill American citizens without a trial, whether the battered leadership of Al Qaeda in Pakistan still poses an imminent threat to Americans, and whether the C.I.A. or the Pentagon ought to be the dominant agency running America’s secret wars.