WASHINGTON -- Republican opposition to U.N. Ambassador Susan Rice as the next secretary of state began to crack Sunday as Sen. John McCain said she was "not the problem" in the White House's handling of the Sept. 11 attack in Libya and suggested he could be persuaded to swing behind her possible nomination.

McCain's comments provide an opening for the Obama administration, which struggled mightily in the weeks leading up to the Nov. 6 election to tamp down speculation of a cover-up involving the attack against the U.S. diplomatic post in Benghazi. The assault killed four Americans, including the U.S. ambassador to Libya.

At issue is Rice's account -- as the administration's representative on the Sunday talk shows Sept. 16 -- that the violence was the spontaneous result of a mob angered by an anti-Muslim video posted on YouTube. She said she relied on talking points provided by the intelligence community that were later discredited.

"I think she deserves the ability and the opportunity to explain herself and her position," McCain, R-Ariz., told "Fox News Sunday." "But she's not the problem. The problem is the president of the United States" who misled the public on terrorist involvement.

McCain's remarks were in contrast to his previous stance that Rice wasn't qualified to replace Hillary Rodham Clinton, who is expected to step down soon as the top U.S. diplomat, and that he would do "whatever is necessary" to block Rice's possible nomination.

Rice is widely seen as Obama's first choice for the job as secretary of State. As the top Republican on the Senate Armed Services Committee, McCain would have considerable sway in the Senate's screening of Rice.

Sen. Lindsey Graham, McCain's close friend and colleague on the committee, told ABC "This Week" he still suspects the White House intentionally glossed over obvious terrorist links in the attack to keep voters from questioning Obama's handling of national security.

But instead of repeating his prior assertion that he was "dead set" against a Rice promotion, Graham suggested he looked forward to hearing her out. If Rice were nominated, "there will be a lot of questions asked of her about this event and others," said Graham, R-S.C.

The subtle shift in GOP tenor on Rice could be the result of internal grumblings on how far to take party opposition. Democrats picked up extra seats in the election to maintain their narrow majority, making it that much harder for the remaining 45 Republicans to block the president's nominees.

One senior GOP Senate aide said Sunday that Republicans hadn't united against Rice and were not convinced that she was worth going after.

"There's a definite sense within the caucus that you have to be conservative about where you put your firepower," said the aide, who spoke on condition of anonymity because the official was not authorized to speak publicly on internal GOP deliberations. "The question is whether the caucus is prepared to filibuster her, and I'm not sure we were."