Talk of Michael Bloomberg running for higher office, which started out as the half-serious project of one of his aides, has officially taken on a life of its own.

This week alone, the Mayor was invited to run for President—unexpectedly, he says—on national television by Nebraska Senator Chuck Hagel, who is himself considering a third-party run for the White House in 2008.

He was also the subject of a sort of draft movement, mounted by a momentary alliance of Albany Republicans and the New York Post, to run for Governor of New York State against Eliot Spitzer in 2010.

In both cases, Mr. Bloomberg’s suitors had ulterior motives: Mr. Hagel needs the Mayor’s money and (maybe) his local popularity to make his insurgent candidacy credible. And State Senate Majority Leader Joe Bruno and the Post—which indulged him with a front-page “Gov Mike!” headline—are using him as a cudgel against their imperious foe, Mr. Spitzer.

But the source of the speculation seems to have become increasingly irrelevant. A story on Tuesday in The Washington Times, citing anonymous sources, said that the Mayor has told associates that he is prepared to spend a billion dollars on a bid for the Presidency.

It hardly even seemed extraordinary.

Mr. Bloomberg, meanwhile, seems at any given moment to be either issuing a ritual denial of the latest rumors or basking in the speculation for all it’s worth.

On May 15, as simultaneous translators at a climate-change conference whispered his words in tinted booths in the back of the Essex House Grand Salon in Manhattan, Mr. Bloomberg took to the role of international statesman with relish.

“The first responsibility of any elected leader is to face the hard questions head-on,” he said, addressing mayors from around the world who had gathered to discuss global warming. “A boyhood hero of mine, former American President Harry Truman, once said that a leader has to lead, or otherwise he has no business being in government.”

After the mayors and their aides had finished eating chicken and cheesecake and listening to Mr. Bloomberg’s speech, they awkwardly paraded out to Central Park South for a press conference. Foreign press photographers jostled to get pictures of the Mayor.

“I didn’t see Bloomberg! I want to see Bloomberg!” said one photographer in Israeli-accented English, in the course of yelling at a colleague to get out of his shot.

Mr. Bloomberg went on to talk about the folly of having an Iraq policy but not an energy policy, discussed relations between state departments (albeit for diplomats paying congestion-pricing tolls), and blamed America’s partisanship for slowing down progress to counter global warming.

“Certainly, in this country, it is partisan politics that has immobilized them,” he said evenly. “To say we can’t start this for a while, until the next election cycle—we don’t have that luxury.”

Mr. Bloomberg’s fellow mayors, at least, seemed taken with idea of him as President.

“I think he would be a very good candidate,” said Beto Richa, the mayor of Curitiba in Brazil.

One woman in attendance asked if he was the “Republican Al Gore.”

In what has become a daily routine, Mr. Bloomberg denied that he was running for President.

Responding to a question about an assertion in the Washington Times story that he had been “meeting with Ross Perot’s most senior people about how they did an independent run in 1992,” the Mayor was casually dismissive.

Mr. Bloomberg said that while he was a neighbor of Mr. Perot and had skied with him a few years ago, “I don’t think I have ever had a conversation with him about national office, and I certainly have never talked with any of his advisors about running for national office.”

His aides, at this point, have taken to asserting that all the speculation is by now self-sustaining. Referring this week to the news-making invitation by Mr. Hagel on Face the Nation, one staffer insisted that none of the Bloomberg people had had any role in engineering it.

“Nobody ever contacted Chuck Hagel,” said the staffer.

The idea, of course, is that it’s all just happening.

Then again, for all his denials, Mr. Bloomberg is hardly doing anything to discourage the attention.

The day before the climate conference that he hosted, Mr. Bloomberg had been up in Albany, again pursuing a conspicuously big agenda—he was there to lobby for his transformative plan for the city, which includes a congestion-pricing measure—and again surrounded by a swirl of speculation about what he might do after (or before) he is scheduled to leave office in 2009.

By coincidence or not, Mr. Bloomberg’s visit to Albany happened on the same day as an unusual press conference to announce a series of endorsements of the Presidential campaign of Hillary Clinton, who is as reliable a draw as any for national political reporters.

“This was a day that worked for everybody,” Mr. Bloomberg told reporters, some of whom debated among themselves whether he was deliberately jumping on the coattails of her attention. “We had this on the calendar before everybody came.”

Mr. Bloomberg—whose prospective Presidential future would be as an independent or third-party candidate—wore a purple tie.

It was the day after the comments by Mr. Hagel, an anti-war Midwesterner who looks like John Wayne—and whom many consider the strongest general-election Presidential candidate the Republicans could field, if only he can get through the conservative-dominated primaries.

“There are people who think he should run for the Presidency,” Mr. Bloomberg said of Mr. Hagel, with the air of someone returning a compliment. “He’s got real ideas. He’s not happy about the same things I’m not happy about.” (He was referring to partisanship in Congress.)

After noting that he has more then 900 days left as Mayor, Mr. Bloomberg referred to his plan for congestion pricing and said, “I’m not running for Governor, and I’m not running for President. The time to do this is now.”

One of the few people in Albany seemingly not taking an interest in Mr. Bloomberg was Mrs. Clinton. After leaving the third-floor Capitol office of Senate Minority Leader Malcolm Smith, she was asked for her thoughts on a possible Presidential challenge by Mr. Bloomberg. Mrs. Clinton took an extra long chug from her bottle of water and ignored the question.

When asked again by the building’s rear doorway, her aides answered for her: “I think we’re done for today.”