Republicans insist the buck stops with the secretary. But although Ms. Sebelius runs the Department of Health and Human Services, the agency directly responsible for the health care law, there are questions about how deeply she was involved in the development of the troubled Web site.

“Kathleen has the title, but she doesn’t have the responsibility or in many respects the kind of wide authority and access to the president that she really needs to make a difference,” said one person close to Ms. Sebelius and the White House, who asked to remain anonymous to discuss internal decision-making. “Everybody thinks that she’s the driving force, but unfortunately she’s not.”

The White House kept close tabs on the creation of the online exchange, with particular attention to the Web site’s design, but managing the details of the software development was left to the Centers for Medicare and Medicaid Services, which Ms. Sebelius oversees. She testified on Capitol Hill this spring that the exchanges were “on track” to open on Oct. 1, but those close to her say she has been far more immersed in developing policy related to the health care law, and in traveling the country to promote it, than in its technical aspects.