In Thursday's hearing, Brennan defended strikes as necessary, saying they are taken only as a "last resort," but he said he had no qualms about the strike that killed U.S.-born cleric Anwar al-Awlaki, because of his roles in several terror attacks.
"The decisions that are made are to take action so that we prevent a future action, so we protect American lives," Brennan said. "That is an inherently executive branch function to determine, and the commander in chief ... has the responsibility to protect the welfare, wellbeing of American citizens."
Still, he said the White House too had considered the concept of the special courts, and he said he would be open to discussing it because "American citizens by definition are due much greater due process than anybody else by dint of their citizenship."
The White House did not offer further comment Friday, and the CIA declined to comment.
Brennan said people are never killed by CIA or military strikes if there is a way to capture them.
Feinstein said at Thursday's hearing that she believed the CIA was open with lawmakers about its part of the program.
"We have provided a lot of oversight over the Predator," she said. "There's a staff team goes out regularly that is at Langley [CIA headquarters] that does look at the intelligence on a regular basis," making more than 30 visits to review strikes and the intelligence leading up to them.
But she said she and senators including Democrats Dick Durbin of Illinois and Pat Leahy of Vermont and Republican Chuck Grassley of Iowa are all looking at the concept of how to regulate the strikes.
On Friday, Republicans were circumspect, with prominent members such as House Intelligence Committee Chairman Mike Rogers of Michigan declining to comment, or reserving judgment until they can see more details.
"I don't know that we can take that exact model and apply it to every tough policy decision that confronts the federal government," said Mac Thornberry of Texas, a member of the House committee. "If someone is shooting at you, you can't go to a court and ask them to shoot back," he said.
Many Democrats were more comfortable with the notion.
"A layer of judicial review could ensure additional checks on the designation of targeted individuals and determine whether sufficient evidence has been produced," said Sen. Mark Udall, D-Colo.
Said Rep. Adam Schiff, D-Calif.: "I don't have in mind to redefine the circumstances in the memo" describing the legal rationale behind the strikes "but rather set up a process for prospective or retrospective analysis of how drone strikes are made,"
The White House allowed lawmakers on the Senate and House intelligence committees to see the classified advice to the president describing the legal rationale behind drone strikes ahead of the Brennan confirmation hearing -- a pre-emptive effort meant to answer increasing questions from lawmakers about the program, and also to head off threatened holds on the Brennan nomination.
But that release has produced further demands for access and information. The intelligence committee members want their staff to read the documents, and the congressional Judiciary committees are also demanding access.