Prominent nominations announced by President Barack Obama and awaiting Senate confirmation include Janet Yellen for chair of the Federal Reserve and Jeh Johnson for secretary of the Homeland Security Department.
There have been a number of Capitol Hill hearings on the Sept. 11, 2012, attack in Benghazi, Libya, that killed Ambassador Chris Stevens and three other Americans. And there have been a review chaired by former Ambassador Thomas Pickering and former Joint Chiefs Chairman Adm. Mike Mullen.
Graham said he wants a joint select committee formed from several individual Senate committees to investigate.
"The State Department is blaming the CIA, the CIA is blaming the State Department. Where was the Department of Defense?" Graham asked earlier Monday on the news show "Fox & Friends."
White House spokesman Jay Carney said the administration had already made "extraordinary efforts to work" with a number of congressional committees investigating what happened before, during and after the Benghazi attacks. "That includes testifying at 13 congressional hearings, participating in 40 staff briefings and providing over 25,000 pages of documents," Carney said.
"Let's be clear that some Republicans are choosing to play politics with this for partisan purposes, and we find that unfortunate," Carney said.
Former Defense Secretary Leon Panetta and Gen. Martin Dempsey, chairman of the Joint Chiefs of Staff, testified for several hours before the Senate Armed Services Committee earlier this year about the military's actions related to Benghazi. Both men answered questions from Graham, a member of the panel.
Under the Constitution, the president nominates but the Senate must approve a host of appointments including ambassadors, Supreme Court justices, Cabinet members and military personnel being promoted to the rank of general officer.
Graham complained on Fox that 14 months after the attack, people who survived it "have not been made available to the U.S. Congress for oversight purposes."
But Jen Psaki, spokeswoman for the State Department, said the department last month sent to the House Oversight Committee a diplomatic security official "who was present in Benghazi the night of the attack." It was the first official confirmation that such a deposition had been given.
She said the committee had issued a subpoena despite warnings that the legal deposition could threaten ongoing law enforcement efforts to bring the perpetrators to justice. A security official based in the Libyan capital, Tripoli, the night of the attack also was made available, she said, adding that more than a dozen State Department officials have testified on the issue.
A blistering report released in December by Pickering, Mullen and three other reviewers found that systematic failures and leadership and management deficiencies at senior levels of the State Department meant security was inadequate for Benghazi and "grossly inadequate" to deal with the attack, in which militants stormed the diplomatic post and set it on fire.
Benghazi stands as cause celebre for conservative Republicans who insist the administration was trying to mislead the American people during the heat of a presidential campaign, playing down a terrorist attack on Obama's watch.