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At the beginning of the Obama presidency, Sen. Lindsey Graham, a reservist in the military’s Judge Advocate General program, had served as a point-man on the administration’s efforts to close Guantanamo Bay, which he and Mr. McCain, and previously President George W. Bush, had agreed needed to be shuttered.

But Mr. Graham, the main architect of the 2006 military commissions law, wanted to couple the closure with comprehensive legislation creating a whole new legal structure to deal with detainees, an effort the White House opposed.

The Obama administration’s failure to work with Congress to clarify detainee laws, Mr. Graham has said, led the military to kill suspected terrorists on the battlefield or through drone strikes, foreclosing the possibility of interrogating the suspects and gleaning crucial intelligence.

Advocates for closing the facility put the onus on Mr. Graham and his colleagues. Because Congress has tied the president’s hands on the issue, they argue, it’s Congress’s responsibility to find a solution beyond simply keeping Guantanamo Bay open and the status of its 171 remaining detainees in limbo.

“It’s been 10 years after 9/11 and it’s getting into a year after Osama bin Laden was killed,” said Heather Hurlburt of the liberal-leaning National Security Network. “We’re going to have to have a conversation about the right ways to protect us now — that Gitmo no longer serves us and what the post-post 9/11 era is going to look like.”