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(CNN) —  

President Barack Obama and his 2008 campaign rival, Sen. John McCain, R-Arizona, are bitter rivals on almost every aspect of U.S. foreign policy. Yet they are allies on one particularly thorny subject – the use of armed drones.

On this highly effective and sometimes imperfect weapon in the war against terrorist groups, the White House wants the Pentagon, not the Central Intelligence Agency, to take the lead in these lethal operations.

The administration’s admission last week that two Western hostages, including American Warren Weinstein, were inadvertently killed in a CIA drone strike in January has prompted key members of Congress to urge a shift in control over the agency’s controversial drone program to the Defense Department.

“We will renew this discussion with the administration, within Congress as to who actually should be running the drone operation,” McCain said on CNN’s “State of the Union” on Sunday.

Asked whether the CIA should be conducting a drone program, McCain responded, “I do not think so. That’s why they are – that’s why they are called the intelligence agency and why we call the armed forces the – obviously, the people that are supposed to be carrying out military operations.”

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In response to the accidental killings of Weinstein and Italian aid worker Giovanni LoPorto, Obama ordered an internal review of the CIA’s drone operations.

A senior administration official indicated Monday the President plans to make a new push to move the CIA’s drone strikes against groups like al Qaeda to the Pentagon.

“The President has explained his belief that we must be more transparent about both the basis of our counterterrorism actions, including lethal operations, and the manner in which they are carried out,” a senior administration official said. “Because of this, he has indicated that he will increasingly turn to our military to take the lead and provide information to the public about our efforts. We continue to work diligently toward this goal,” the official added.

The White House attempted to transfer much of the CIA program over to the Defense Department two years ago, arguing the Defense Department already conducts its own drone strikes against more traditional military targets.

“He believes very much in the need to be as transparent as possible on these matters with Congress as well as with the public,” then-White House Press Secretary Jay Carney told reporters at the time.

During a speech at the National Defense University in 2013, Obama called on Congress to support his efforts for greater transparency in U.S. drone operations. He responded directly to a protester who interrupted his address to demand an end to the intelligence agency’s armed UAV operations.

READ: Will the drone debate come back?

“Can you take the drones out of the hands of the CIA? Can you stop the signature strikes killing people on the basis of suspicious activities,” the demonstrator asked. “We’re addressing that, ma’am,

Obama answered.

But congressional opponents from both parties, including leaders from the House and Senate Intelligence Committees, opposed the move.

“Turf battles” is how McCain explained that tug-of-war on CNN Sunday. But the Republican senator insisted now that drones have become a primary weapon against terrorists, the military should operate the program.

“And I can understand, when it was a very small operation, why it would be done by the intelligence agencies, such as U-2s and other reconnaissance aircraft, for many years. Now it has reached the point where it’s an integral part of the conflict,” McCain said.

Under the administration’s renewed plan for the Pentagon to have greater control over drone operations, the CIA would continue to have a role, providing intelligence and expertise. But as in the case of the last White House drone push, administration officials will face a challenge in seeking congressional support.

Still, McCain is not the only Republican who believes CIA drone strikes should end. Ohio’s GOP governor John Kasich, a potential presidential candidate, has also called for the program to be shifted to the Defense Department.

“I don’t believe the drone program ought to be run out of the CIA,” Kasich said on CNN’s State of the Union. “The CIA is an intelligence-gathering operation. The operation, the drone program, should be operated exclusively out of the Pentagon,” he added.