In hindsight, of course, that confidence was misplaced. “It’s not that they took their eye off the ball — they spent a lot of time on it,” said John D. Podesta, who helped run Mr. Obama’s transition and remains close to the White House. “The question is whether they spent their time on the right things. It wasn’t centralized. You can decide that was Obama’s problem or Denis’s problem or Kathleen’s problem. But it was a problem.”

Mr. McDonough’s defenders say the real mistakes were made before he took over, when the White House put its policy officials rather than outside experts in charge of setting up the major web operation for the health care exchanges. Mr. Obama’s ill-founded promise that any Americans who liked their health plans could keep them likewise preceded Mr. McDonough’s tenure.

But Mr. McDonough was in the corner office when it all went awry. “It’s the age-old thing: Because you’re the chief of staff, you’re responsible for it,” said William M. Daley, another of Mr. McDonough’s predecessors. “Whether it’s legitimate or not, a lot of people would say it happened on your watch, so you’re responsible for it. At some point you’ve got to do a debrief on how did this thing really get screwed up — who was debriefing who, did we just not see this, how did we miss it?”

Colleagues say Mr. McDonough seems determined to figure out how his methodical structure broke down and what could have been done differently, but only after the problems are fixed. On the surface, at least, he exudes the calm of an aide who has been through a lifetime of crises over the past five years.

“When you see Denis, you don’t get the sense that his hair is on fire,” said Steve Elmendorf, a Democratic strategist who has advised the White House. But “he understands if they don’t fix it, it’s unacceptable.”

So far, virtually no one has called for Mr. McDonough’s head, and it is a measure of the credibility he has built up that he has not come under more fire. The political barbs from the outside — and the private recriminations from the inside — are aimed more at Ms. Sebelius. “He’s ready to hear bad news, so people never think he’s part of the problem,” said Senator Charles E. Schumer, Democrat of New York.

Mr. McDonough has also benefited by tending to Republicans and returning phone calls that once went unanswered. “I’ve been in Washington for 14 years, and I saw more of him in the eight weeks of budget and deficit discussions than I saw all the other presidents’ chiefs of staff combined,” said Senator Johnny Isakson, Republican of Georgia. “And never had one give me his home number.”