After the troubles with Mr. Daschle and other nominees, the White House has intensified its vetting to make sure it thoroughly scrubs its choices before Mr. Obama signs off.

Ms. Sebelius’s family and her administration have both been notably bipartisan. Her father was a Democratic governor of Ohio and her father-in-law was a longtime Republican member of Congress from Kansas. She won her first term with a former Republican businessman as her running mate and a second term with the former Republican state party chairman on her ticket.

Ms. Sebelius became an early supporter of Mr. Obama by endorsing him in January 2008 and was mentioned as a possible vice-presidential nominee. After the election, her name was floated for a variety of possible jobs, including secretary of labor, agriculture or education.

If she becomes health secretary, she will be the fourth woman in the 15-member cabinet, overseeing 65,000 employees and a $700 billion budget. But she is unlikely to also wear the second hat that Mr. Daschle negotiated for himself as White House health czar, a position that could be influential in setting health care policy.

Image Gov. Kathleen Sebelius in her office in Topeka, Kan., on Feb. 9. Credit... Orlin Wagner/Associated Press

Administration officials said that was a special arrangement for Mr. Daschle, adding that they were still considering what to do about that job.

Should Ms. Sebelius get the cabinet post, she would be replaced as governor by Lt. Gov. Mark Parkinson, the former Republican chairman who switched to the Democratic Party when he ran with Ms. Sebelius in 2006. Mr. Parkinson has said he will not run for governor in 2010.