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Mr. Graham made clear Wednesday that Mrs. Rice has not been the sole recipient of his broadsides, pointing out on the radio show that he asked Mr. Clapper to resign in March because he lacked confidence in his ability to handle national security, and he asked counterterrorism chief John O. Brennan to resign “when he said that a 20 percent recidivism rate coming out of Gitmo was acceptable,” a reference to the U.S. detention camp at Guantanamo Bay, Cuba.

“I think it’s a good thing for people of color and women to be appointed to the highest level of our government, because in the past they couldn’t be,” Mr. Graham said. “I think it would be a terrible thing in America if you couldn’t question and challenge what they did in their jobs.”

Rep. James E. Clyburn, the third-ranking Democrat in the House, hinted Tuesday that there are racial undertones to criticism of Mrs. Rice.

Rep. Marcia L. Fudge, Ohio Democrat and incoming chairwoman of the Congressional Black Caucus, has made similar comments.

“It is a shame that anytime something goes wrong, they pick on women and minorities,” she said. “I have a real issue with that.”

Rep. Michael C. Burgess of Texas, one of 97 House Republicans who signed a letter to Mr. Obama sent this week opposing Mrs. Rice’s possible nomination for secretary of state, called Mr. Clyburn’s comments “absolutely false.”

“I don’t know where … Rep. Clyburn gets that, his information,” Mr. Burgess said Wednesday on CNN’s “Starting Point.” “I will just tell you he is factually incorrect.”