The proposal is under "serious consideration," a US official said.
The official said no final decision has been made, and that there is no specific time frame in place, but that the change is being considered "due to a desire for greater transparency in who is being targeted."
By law, the military is not able to act covertly the way the CIA can, and it must answer to Congress.
The shift would happen over time.
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"It won't happen overnight," the US official said, adding that operations in some countries would transfer to US military control before other nations.
"Yemen is an example of one of the first programmes that could shift," the official said, while shifting the responsibility for the programme that operates in Pakistan "would be much further out."
CIA Director John Brennan has expressed a desire to move the intelligence agency back to traditional intelligence-collecting.
The transfer is being contemplated at a time when US drone attacks in Pakistan, Yemen and elsewhere have been facing increasing scrutiny and questions from human rights groups who say that innocent lives are being lost because of such covert operations.