Some lawmakers have suggested the White House attempted to cover up the circumstances of the attack by saying it was spontaneous.
Administration officials confirmed last week the consulate was essentially a front for a much larger CIA base about a mile away. Most of the 30 Americans evacuated after the attack were CIA employees or contractors, not diplomats.
"We want to probe everybody who was involved, all the way up to and including the president of the United States," McCain said.
Five of the eight Republican members of the Senate Homeland Security and Governmental Affairs Committee -- including McCain -- did not attend a classified briefing on the attack, conducted Wednesday by administration officials, CNN reported.
CNN said one of its reporters approached McCain at the Capitol Thursday to ask why he did not attend, but McCain declined to answer, saying, "I have no comment about my schedule and I'm not going to comment on how I spend my time to the media."
When the reporter asked McCain why he would not comment, McCain said, "Because I have the right as a senator to have no comment and who the hell are you to tell me I can or not?"
A spokesman for McCain subsequently told CNN in an email the senator missed the briefing due to a "scheduling error."
CNN noted McCain was holding a news conference at noon Wednesday while his committee was being briefed on the Benghazi attack.
One senator who did attend the briefing, Connecticut independent Joe Lieberman, said afterward he thinks the Homeland Security Committee can do a proper job of investigating Benghazi, Politico reported.
The top Republican on the committee, Sen. Susan Collins of Maine, concurred.
"I do not see the benefit of creating a brand new committee when we already have the Senate's chief oversight committee, plus the Intelligence Committee, examining this very important matter," Collins told Politico.
First-term Republican Sen. Marco Rubio of Florida Thursday distanced himself from the comments of McCain and Graham on whether Susan Rice, the U.S. Ambassador to the United Nations, was disqualified from consideration as a successor to U.S. Secretary of State Hillary Clinton.
Rubio said President Barack Obama has the right to choose whoever he believes will most help the country and the Senate should wait to respond until a nominee is chosen.
Petraeus, who resigned as CIA director Friday over an extramarital affair, agreed to testify before the House and Senate intelligence committees on the Benghazi attack, lawmakers said.
The former four-star general -- who ran the CIA for 14 months, including during the Benghazi attack -- was to testify before the Senate Intelligence Committee in a closed-door session Thursday or Friday, panel Vice Chairman Saxby Chambliss, R-Ga., told several news organizations Wednesday.
The hearing would not address his affair, committee Chairwoman Dianne Feinstein, D-Calif., said.
Petraeus is to testify separately behind closed doors before the House Intelligence Committee Friday, the committee said Wednesday evening.
Petraeus' successor -- acting CIA Director Michael Morell -- was to testify in private before the House and Senate intelligence committees Thursday. Other top-ranking intelligence officials scheduled to testify included National Intelligence Director James Clapper, Petraeus' boss.
The House Foreign Affairs Committee was to hold an open hearing on the Benghazi attack beginning at 10 a.m. Thursday.
The Senate Foreign Relations Committee held a closed briefing Tuesday.
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