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

Image Kathleen Sebelius Credit... Michael Reynolds/European Pressphoto Agency

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. Democrats have not held a Senate seat in the state since 1939. And even before the president’s popularity started to take a steep slide last year, he fared especially poorly in Kansas, winning only 38 percent of the vote there in 2012.

Democrats say that Ms. Sebelius would be their best hope at winning in a tough state, especially if Mr. Roberts loses his primary to Milton Wolf, a Tea Party-backed radiologist who has alarmed mainstream Republicans with some of his actions, such as when he posted gruesome pictures of gunshot victims on Facebook.

Perhaps more significant, Ms. Sebelius would force Republicans to spend money in Kansas as they tried to fight off her challenge. Her family has a long history in the state, and she was a popular, twice-elected governor. In 2006, she was re-elected with 58 percent of the vote.

But friends and Democrats who know her said that they seriously doubted she would follow through and mount a campaign, considering how personally difficult the past six or seven months had been. She has been mocked and excoriated by Republicans on Capitol Hill, who made her the face of the Affordable Care Act’s problems. And she has told people that she anticipated staying on in her job as secretary for a while longer. President Obama has already nominated a successor, Sylvia Mathews Burwell, the current director of the Office of Management and Budget, but her appointment must be approved by the Senate.