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Mr. McCain and Mr. Graham earlier Wednesday said they would do “whatever is necessary” — including a filibuster — to stop Mrs. Rice from becoming the next secretary of state if Mr. Obama does pick her.

Sen. John F. Kerry, Massachusetts Democrat, is also reportedly under consideration for Mrs. Clinton’s job.

Mr. Graham told The Times that he doesn’t have any concerns about Mr. Kerry — “If the president thought that John Kerry could serve his administration and country, I’d be very inclined to support him,” he said.

He added that Mrs. Rice is a single nomination, not an indication of broad opposition to other nominees.

But his idea for a select committee to investigate Benghazi is meeting with resistance from top Democrats, who said there are multiple committees already looking into the run-up to the attack and the slow U.S. military response.

Ambassador J. Christopher Stevens, who was one of four Americans killed in the attack, had asked for more security, but his request was denied. There were several warnings of increased terrorist activity, including an attack on the consulate in June, which Mr. Graham said Mr. Obama must answer whether he knew about.

A Defense Department timeline released last week showed the military took hours to get a security team to the site of the attack.

Mr. Obama in his news conference defended his own handling of the attack.

“I can tell you that immediately upon finding out that our folks were in danger, that my orders to my national security team were do whatever we need to do to make sure they’re safe,” he said.

When pressed during Wednesday’s news conference to provide more answers to the families of the victims of the Benghazi attacks, Mr. Obama said he would address the families directly, not through the press. But he also referred to an ongoing investigation, saying he and the administration would “provide all the information that is available about what happened that day.”

When it comes to the bloody civil war in Syria, Mr. Obama said that he is encouraged that the Syria opposition has formed a new, more representative leadership council, but that the U.S., unlike some other countries, isn’t ready to recognize the group as a “government in exile” or to arm it.

The president also pledged to try to engage Iran in negotiations over its disputed nuclear program, saying he wouldn’t be “constrained by diplomatic niceties and protocols” even though he believes there is “still a window of time for us to resolve this diplomatically.”

David Sherfinski contributed to this report.