GOP Sens. John McCain (Ariz.), Lindsey Graham (S.C.) and Kelly Ayotte (N.H.) issued the thinly veiled threat against the Brennan nomination during a press conference on Capitol Hill on Thursday.
"That will be a subject that will be addressed" as the Senate considered Brennan's confirmation to become the nation's top spy, McCain told reporters.
A week after the attack, which ended with the death of four Americans, including Ambassador Chris Stevens, administration officials blamed the strike on anti-American protesters.
Weeks later, White House leaders admitted the consulate attack was a coordinated assault by an al Qaeda-affiliated terror groups based in Benghazi.
When asked specifically if he or other Senate Republicans would filibuster the Brennan nomination if he did not name names regarding the administration's Benghazi story, McCain replied sharply: "We are going to get that information."
Graham had a more optimistic outlook when asked if the Obama administration would meet lawmakers' demands on Benghazi, saying coyly: "Hope springs eternal."
That said, McCain was confident he would get what he wanted from the administration, given that White House officials "have already been forthcoming" on prior requests regarding the Brennan nomination.
The White House acquiesced to lawmakers' demands over the Brennan nomination last Wednesday when it opted to release classified memos on the CIA and Pentagon's armed drone program to the Senate Intelligence Committee, prior to Brennan's confirmation hearing.
But on Wednesday, Intelligence Chairwoman Dianne Feinstein decided to postpone the panel's confirmation vote on Brennan, after committee members requested more information on Benghazi and the armed drone program.
The California Democrat did ask for more information on the use of armed drones against terror suspects, particularly American citizens, but did not request details on Benghazi, according to a congressional source.
Wednesday's Republican threats over the Brennan bid came a day after the White House admitted Obama did not call his Libyan counterpart the night of the deadly attack on in Benghazi.
In a letter to McCain, Graham and Ayotte sent Tuesday, White House counsel Kathryn Ruemmler said it was then-Secretary of State Hillary Clinton who called President Mohammed Magariaf on Obama's behalf that night “to coordinate additional support to protect Americans in Libya and access to Libyan territory."
The president then called Magariaf the next evening, Ruemmler added.
That admission proved "the president was virtually disengaged" from "an attack you could see coming," Graham said on Thursday.
That lack of a response represented a "breakdown in national security," according to Graham, who added that if Obama had called the embassy in Tripoli or anyone else in Libya that night, "it could have made a difference."
Brennan had been expected to clear the committee and be confirmed by the Senate despite the controversy over the Obama administration's drone strike program on American terrorist suspects overseas.
On Tuesday, Brennan testified during a classified session before Feinstein's panel on efforts to thwart al Qaeda and its affiliates across the globe, particularly the group's increasing presence in North and West Africa.
The closed hearing was seemingly the final hurdle in the White House's plan to have Brennan become the head of CIA.