But in an interview, Ms. Sebelius’s older brother, Donald D. Gilligan, said, “I don’t think you resign in the middle of a fight.”

Moreover, Mr. Gilligan said: “The fact that people are calling for her head does not surprise her or alarm her particularly. People have been calling for her head for a long time, and it’s still there. I don’t think that fazes her much.”

Mr. Gilligan said Ms. Sebelius was “just digging in on a daily basis and trying to figure out how to fix” the problems that have riddled the insurance exchange.

The secretary’s sister, Ellen M. Gilligan, said, “To my knowledge, she is not going to resign,” despite the “wishful thinking” of some Republicans.

“She serves at the pleasure of the president,” she said. “If the president decides that it is her responsibility and that she is ultimately responsible for the failure of the exchanges, then, of course, she will resign.”

But Ms. Gilligan said that was unlikely.