For someone with an elevated position in Al Qaeda, Mr. Shami has kept an unusually low profile. He has made no propaganda videos, nor does he seem to have been mentioned in any of the myriad online forums that militant groups use to motivate their followers, raise money and recruit new fighters.

The SITE Intelligence Group, which monitors the online communications and propaganda videos of militant groups, said it had no records of Mr. Shami’s name ever being discussed in chat rooms.

Some terrorism experts said that it could be a calculated strategy by Al Qaeda not to broadcast the growing role of an American inside the organization, or that the group might refer to Mr. Shami by another name. Still others said that it might just be an act of self-preservation on Mr. Shami’s part, given that so many of the people who have planned external operations for Al Qaeda — including Ilyas Kashmiri and Saleh al-Somali — were killed by American drones.

“At some point, it would probably make sense not to advertise these positions,” said Seth Jones, a terrorism expert at the RAND Corporation. Mr. Jones said that Mr. Shami had made “a very conscious effort to stay below the radar.”

The debate over Mr. Shami’s fate is the first time that the Obama administration has discussed killing an American citizen abroad since Anwar al-Awlaki was killed in a C.I.A. drone strike in Yemen in September 2011. It comes less than a year after Mr. Obama announced new guidelines to tighten the rules for carrying out lethal drone operations. When the president announced the guidelines, during a speech in May in Washington, the White House acknowledged that four American citizens had been killed in drone strikes during Mr. Obama’s time in office.

According to the White House, only Mr. Awlaki had been targeted.

As it was in Mr. Awlaki’s case, the Justice Department has been enlisted to evaluate whether a lethal operation against Mr. Shami is legally justified, but it appears that the Obama administration remains divided on the issue. Several officials said that the C.I.A. has long advocated killing Mr. Shami, and that the Pentagon, while initially reluctant to put him on a target list, has more recently come to the C.I.A.’s position.

It is unclear what Mr. Obama’s position is on whether Mr. Shami should be targeted. American officials said that as part of the new rules ordered by Mr. Obama, the Pentagon, rather than the C.I.A., is supposed to carry out any lethal strike against an American overseas, a provision intended to allow government officials to speak more freely about the operation after it is carried out.