“There’s no question that from a very limited start, she mastered an incredible amount of material, became quite expert and quite sophisticated,” said Bruce Vladeck, who was Mr. Clinton’s Medicare chief. “Maybe this is the story of her life. Now that health care is achievable, that the political moon and stars are lined up to do it, she’s no longer there.”

Image President Obama hopes to learn from protests to the 1994 plan. Credit... Gary Stewart/Associated Press

Some who know Mrs. Clinton predict she will not be able to stay entirely out of the debate. Among them is Senator Max Baucus, the Montana Democrat and chairman of the Finance Committee, who said he last spoke with Mrs. Clinton about health care before she took her cabinet post.

“Oh, Hillary’s very smart, she’ll figure out a way to sidebar with the president,” Mr. Baucus said. “I have a hunch that she’ll give a word or two of advice to the president and to me, and I intend to talk to her.”

Others, like Donna E. Shalala, who was health secretary to Mr. Clinton, say Mrs. Clinton does not need to advise Mr. Obama on the lessons of the past; she already did so when she ran for president by laying out a plan far different than the complicated proposal, heavy on government intervention, that flopped so badly when she was first lady.

“The way she laid out the issue during the campaign is a clear indication that they learned something over the last decade, and her proposal was very close to Senator Obama’s,” Ms. Shalala said. “That reflects what we learned  that people don’t want to lose the health insurance they currently have, that they are desperately concerned about cost, that they want evidence based medicine and prevention. I think in many ways she’s done her work.”

For Mr. Emanuel and other Clinton veterans who now work for Mr. Obama  among them Lawrence H. Summers, the chief economics adviser; Peter R. Orszag, the budget director; and Neera Tanden, a longtime policy aide to Mrs. Clinton who is now a senior official at the Department of Health and Human Services  Mr. Obama’s effort to remake the nation’s health system is something of a do-over.

It began with the economic recovery bill Mr. Obama recently signed, a measure that included $20 billion to modernize medical records and $1.1 billion for research on the cost effectiveness of drugs and medical treatments, initiatives that the White House says are critical to helping to bring down the cost of care.