Ms. Sebelius is considering entreaties from Democrats who want her to run against that old friend, Senator Pat Roberts, Republican of Kansas.

Several Democrats said this week that Ms. Sebelius had been mentioned with growing frequency as someone who could wage a serious challenge to Mr. Roberts, 77, who is running for a fourth term and is considered vulnerable. One person who spoke directly to Ms. Sebelius said that she was thinking about it, but added that it was too soon to say how seriously she was taking the idea.

It was only last week, after all, that Ms. Sebelius, 65, said that she would step down from her cabinet job.

Even if Ms. Sebelius had not presided over the Department of Health and Human Services at a time of turmoil and self-inflicted distress — and while carrying out a law that inspires such anger on the right — her candidacy would be a tough sell in Kansas.