WASHINGTON — White House officials, asserting that the HealthCare.gov website is largely fixed, are under mounting pressure from Democrats and close allies to hold senior-level people accountable for the botched rollout of President Obama’s signature domestic achievement and to determine who should be fired.

For weeks, the president and his aides have said they are not interested in conducting a witch hunt in the middle of the effort to rescue the website. But in the West Wing, the desire for an explanation about how an administration that prides itself on competence bungled so badly remains an urgent mission.

“I assure you that I’ve been asking a lot of questions about that,” Mr. Obama said in a news conference last month, in comments that reverberated across the administration. The president warned, “There is going to be a lot of evaluation of how we got to this point.”

Officials declined to offer details about which government employees at the White House or other agencies might be under the microscope during any review of the development of the health care website. But there is a long list of people who have been publicly identified as key players. The possible targets include Kathleen Sebelius, the health and human services secretary; Marilyn Tavenner, the head of the Centers for Medicaid and Medicare Services; Mike Hash, the head of the health and human services health reform office; Michelle Snyder, the chief operating officer at Medicaid and Medicare; Henry Chao, the chief digital architect for the website; Jeanne Lambrew, the head of health care policy inside the White House; David Simas, a key adviser involved in the rollout; and Todd Park, the president’s top adviser on technology issues.