NEW YORK » Mayor Michael R. Bloomberg has long struggled to imagine a successor with the combination of star power, experience and grit to fill his shoes.

But not long ago, he was struck by an inspiration: Hillary Rodham Clinton, the retiring secretary of state.

In a telephone call confirmed by three people, Bloomberg encouraged Clinton to consider entering the 2013 mayor’s race, trading international diplomacy for municipal management on the grandest scale.

She would, the mayor suggested, be a perfect fit.

Much about the call, which occurred some months ago, remains shrouded in mystery. But Bloomberg’s overture to the former first lady highlights the level of his anxiety about the current crop of candidates, his eagerness to recruit a replacement who can rival his stature and his determination to become a kingmaker in the political arena he will soon exit.

In Clinton, it seems, a mayor known for his sometimes unsparing critiques of those in public life sees a globe-trotting problem solver like himself.

During their conversation, Clinton left little room for doubt: She was not interested in seeking the mayoralty, people briefed on the call said.

Even so, Bloomberg’s reaching out to her is rich with political intrigue. He has privately signaled support for the presumptive candidacy of Christine C. Quinn, the speaker of the City Council, who cleared a path for his third term by backing a change to the city’s term-limit law. The mayor’s political apparatus has begun to coalesce around Quinn, and she promotes herself as a leader who ran the city alongside Bloomberg.

In many corners of the city’s political world, Bloomberg’s eventual endorsement of Quinn has been considered a foregone conclusion, barring the entry of a big-name candidate into the field.

But the mayor’s conversation with Clinton, even after he had begun to telegraph his allegiance to Quinn, suggests that, to a degree previously unknown, his thinking has been unsettled.

"He is looking for somebody he can feel comfortable handing the reins over to," said Hank Sheinkopf, a longtime New York City political operative who worked on Bloomberg’s last campaign.

Spokesmen for both Bloomberg and Clinton declined to comment, saying that, as a rule, they do not discuss private conservations.

The people familiar with Bloomberg’s conversation with Clinton characterized it as a casual but earnest discussion that touched on the futures of them both.

Bloomberg maintains close and warm relations with Clinton, whose ascent within the New York political scene, in the early 2000s, closely tracked his own: Her tenure as a U.S. senator overlapped with Bloomberg’s mayoralty for seven years, a period when they spoke frequently about state and federal policy and finances.

Those familiar with their exchange over the mayor’s race say Bloomberg holds Clinton in extremely high regard, admiring her willingness to devote four grinding years to working for a former rival, President Barack Obama, and her effectiveness as a diplomat. (In 2008, he invited the Clintons to drop the crystal-studded ball in Times Square on New Year’s Eve, standing beside them on stage.)

He also relishes the chance to the course of future campaigns, in New York and beyond, a goal he has started to institutionalize recently with the formation of his own super PAC.

Despite his stated allergy to decision-making by poll, Bloomberg is well aware of Clinton’s standing as a singularly popular Democrat who would upend the mayoral campaign.

Clinton is not a resident of New York City, a requirement for incoming mayors. But she has quickly overcome such hurdles in the past. To establish New York state residency for her 2000 run for Senate, she and her husband bought a home in Chappaqua, N.Y.

Democratic operatives and fundraisers said the prospect of Clinton’s seeking citywide office struck them as improbable, whether or not she ever mounts a second run for the White House.

As much as anything, they said, Bloomberg’s encouragement seemed to reflect his lofty view of the office — and himself.

"If in it fact he did say that to Hillary Clinton, it’s only because he holds the position and therefore regards it as a step up from being president," said Robert Zimmerman, a Democratic National Committee member from New York and a longtime fundraiser for the party.

The city’s business community, of which Bloomberg is a leading member, has long fretted that after his 12-year mayoralty concludes, leadership of the city is likely to return to an elected official with little or no boardroom experience.

Clinton hardly fits the description of corporate chief executive — though she briefly served on the board of Wal-Mart Stores — but she has managed a large government bureaucracy as secretary of state, overseeing tens of thousands of employees in scores of countries.

In contemplating Clinton’s future, Bloomberg was indirectly dealing with a challenge that has endlessly fascinated and bedeviled him: how to conduct a vibrant post-government life.

Bloomberg postponed his own version of that discussion three years ago by seeking a third term. Now he is becoming steeped in it, telling friends and colleagues that he is not entirely sure how he will spend his time after Dec. 31, 2013, when he must step down.

© 2012 The New York Times Company