Until now, the Pentagon has had the lead role for conducting airstrikes — with drones or other aircraft — against militants in Afghanistan and other conflict zones, such as Somalia and Libya and, to some extent, Yemen. The military publicly acknowledges its strikes, unlike the C.I.A., which for roughly a decade has carried out its own campaign of covert drone strikes in Pakistan that were not acknowledged by either country, a condition that Pakistan’s government has long insisted on.

But the C.I.A.’s director, Mike Pompeo, has made a forceful case to Mr. Trump in recent weeks that the Obama-era arrangement needlessly limited the United States’ ability to conduct counterterrorism operations, according to the current and former officials, who would not be named discussing internal debates about sensitive information. He has publicly suggested that Mr. Trump favors granting the C.I.A. greater authorities to go after militants, though he has been vague about specifics, nearly all of which are classified.