WASHINGTON (Reuters) - Republican U.S. lawmakers turned up the heat on Sunday on Susan Rice, saying the U.N. ambassador - seen as a possible nominee to replace Hillary Clinton as secretary of state - must testify before Congress on her remarks after the September attack that killed the American envoy to Libya.

Two influential Senate Republicans, John McCain and Lindsey Graham, did not back down on Sunday from their vow made last week to oppose any attempt by President Barack Obama to put Rice into a Cabinet position that would require Senate confirmation.

"She has a lot of explaining to do. I am curious why she has not repudiated these remarks," McCain, the top Republican on the Senate Armed Services committee, said on CBS's "Face the Nation" program.

Obama last Thursday warned Republicans that if they had a problem with the U.S. handling of the Benghazi attack in Libya to "go after me" rather than picking on Rice.

McCain said he wished the president would not waste time getting mad at him but instead spend the time finding out what happened in Libya and how could it be prevented in the future.

"She's going to have to come in and testify at some point, whether it's in a closed hearing or an open hearing," Republican Saxby Chambliss, vice chairman of the Senate Intelligence Committee, told the "Fox News Sunday" program, referring to Rice.

"We're going to have an open hearing, too. But at some point, she needs to come in and say what the president or the White House directed her to say," Chambliss added.

Republicans have criticized Rice for appearing on Sunday morning news shows five days after the September 11 attack on the U.S. diplomatic mission in Benghazi and saying that preliminary information suggested the assault was the result of protests over an anti-Muslim film rather than a premeditated strike.

The U.S. ambassador to Libya, Christopher Stevens, and three other Americans were killed in an attack that has raised questions about the security of the diplomatic mission, U.S. intelligence about the threat, and the adequacy of the immediate U.S. response.

The White House has said Rice's comments were based on the best information Rice had at the time. Republicans have used her early assessment as a cudgel for criticizing the administration as not being forthcoming about Benghazi. The senators' remarks last week suggested they would pursue the issue even though the U.S. presidential election is over.

'DESTROYS THE NARRATIVE'

Graham said Rice's initial explanation helped reinforce the Obama administration's "narrative" that al Qaeda has been weakened.

"Had the truth come out a few weeks before the election that our consulate in Benghazi had been overrun by an al Qaeda-sponsored, affiliated militia, that destroys the narrative we have been hearing for months that al Qaeda's been dismantled," Graham said on NBC's "Meet the Press" show.

"The truth of the matter is nothing could be further from the truth, and the story she told helped reinforce the political narrative helpful to the president," Graham said. "I don't know what she knew, but I know the story she told was misleading. I don't know why it was misleading."

Asked if Rice has a chance of being confirmed by the Senate to another post, Graham said he would listen to what she had to say but he was "very disappointed in Susan Rice."

When McCain was asked if he would support Rice if she were nominated as secretary of state, McCain said: "Under the present circumstances, until we find out all the information as to what happened, I don't think you could want to support any nominee right now."

"This is very, very serious, and it has even larger implications than the deaths of four Americans," McCain added. "It goes right to the heart of the 'light footprint' policy that this administration has been pursuing and all of the failures throughout the Middle East."

Lawmakers appearing on the Sunday shows expressed no appetite for the proposal by McCain and Graham for the creation of a special congressional committee to investigate the Benghazi attack, rather than have the existing committees with jurisdiction hold hearings.

"The committees within the United States Senate are very capable of investigating this in the right way," Chambliss said.

Senator Joseph Lieberman, an independent who has worked closely with McCain and Graham, added that he disagreed with "my two amigos" on the matter and that "our committees can handle this and come up with the answers."

Senator Carl Levin, the Democratic chairman of the Senate Armed Services Committee, called the criticism of Rice "one of the most unfair attacks I've ever seen in Washington in 34 years. Susan Rice was using the unclassified talking points which were provided by the intelligence community."

"The issue is whether or not Susan Rice should be pilloried for using a intelligence report which (former CIA chief) David Petraeus signed off on, which the DNI, the director of national intelligence, Mr. (James) Clapper, signed off on," Levin said.