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But behind the scenes, a different reality is playing out.

High-powered advisers to Mr. Bloomberg — and even the mayor himself — have chewed over alternatives, joked about dream candidates and even floated the possibility of a mayoral run to at least five boldface figures, highlighting their worry that City Hall could fall into less nimble hands.

The conversations have occurred over dinners and by telephone, in tones both serious and playful. The prospects span the world of government and business, a group whose members dwell within the five boroughs and beyond the city's borders.

Mr. Bloomberg has mused about a Mayor Charles E. Schumer with the Democratic senator from New York, and teased Mortimer B. Zuckerman, a fellow billionaire media mogul, about a possible bid. The mayor's advisers raised the idea of a run with Edward G. Rendell, the former Democratic governor of Pennsylvania and mayor of Philadelphia, and with Edward Skyler, Mr. Bloomberg's former top deputy in City Hall, according to several people.

The mayor's most formal overture was delivered to Secretary of State Hillary Rodham Clinton, perhaps Mr. Bloomberg's most quixotic choice for the job. The mayor personally encouraged her to enter the race about a year ago, three people who were told about the discussions have told The New York Times.

The conversations suggest that Mr. Bloomberg and his aides long for somebody who can match his own blend of celebrity, success and self-assurance.

"The mayor believes he is special," Mr. Rendell said in an interview in his Philadelphia office. "He wanted somebody at a very high level to come in to do a job he has often said to me — and he's not the only person who says it — is the second most difficult job in the country."

A spokesman for Ms. Quinn's campaign, Josh Isay, declined to comment.

The discussions involving the mayor and his team — the oldest occurring about a year ago, the most recent after Hurricane Sandy — were described by people with knowledge of the talks. Many of these people spoke on the condition of anonymity for fear of upsetting those involved or breaching confidences.

Howard Wolfson, a deputy mayor who oversees Mr. Bloomberg's political operations, denied the claims that the mayor had encouraged either Mr. Schumer or Mr. Zuckerman to enter the race.

A spokesman for Mr. Schumer declined to comment.

Mr. Zuckerman said in an interview that Mr. Bloomberg had spoken with him about running.  "A lot of people have talked to me about that possibility," Mr. Zuckerman said. "He's not the only person."

The flirtations are unwelcome news for Ms. Quinn, who has been Mr. Bloomberg's reliable partner in city government for years. After the revelations about Ms. Clinton in December, the mayor hastened to publicly affirm his admiration for the speaker, saying at a news conference that she had "made an enormous difference in this city."

Not all the overtures appeared practical.

Mr. Schumer, elected to the Senate in 1998, has told associates he is a born legislator, and he has passed up chances to run for other executive offices, like governor, in the past.

Mr. Rendell lives about 100 miles away in Pennsylvania and would have to move to New York City to run. Despite having grown up in New York, he said he knew little of the city beyond Manhattan. "I'm not sure how many times I've stepped foot in Brooklyn," he said. "I have no understanding of Queens and no understanding of the Bronx." Running for mayor of New York, he said, "didn't sit well with me."

The suggestion to Mr. Rendell came from a longtime Bloomberg adviser, Bradley Tusk, even though Mr. Tusk is now informally advising Ms. Quinn. Mr. Tusk, a former Rendell intern who ran Mr. Bloomberg's 2009 campaign, said the impetus to approach Mr. Rendell "came from a few people in the business community, and not anyone remotely connected with Mayor Bloomberg."

None of the figures involved plans to enter the race, but several conceded that the interest from the Bloomberg circle had prompted them at least to consider the idea, however fleetingly.

"I would love to be in that job," said Mr. Zuckerman, a student of policy who has no party affiliation and weighed running for the Senate a few years ago.

He insisted that Mr. Bloomberg's suggestion had an informal "teasing" feel, even as he acknowledged a longstanding call to public service in New York.

"If I could be appointed, I'd probably be serious about it," he added, wryly.