The 2004 presidential race lacked a feature present in every campaign since 1992: a significant showing by a third-party candidate. Today, however, political strategists are coming to recognize that the presidential race in 2008 may be a threesome again. Mayor Bloomberg may have repeatedly denied that he is running, but after two terms as New York mayor, what else would he do with his money? “Half a billion dollars? Not a problem,” he was reported to have said earlier this year when asked what he was willing to spend on a campaign.

The prospect of an independent forcing a three-way split is not appetizing to the mainstream candidates. “There’s been discussion about Bloomberg, and no one likes the idea of a third party getting in the race with millions and millions of dollars,” an adviser to one of the leading Republican contenders said. “If the Republican candidate ended up being a real firebrand like” Senator Brownback of Kansas, “I can see Bloomberg carving out some territory.”

A veteran campaigner for President Bush and a current adviser to Senator McCain, Mark McKinnon, told New York magazine that if the primaries “squeeze out the moderates, you’ll have an ideal situation for a third-party run.”

The Internet guru credited with transforming Howard Dean from an unknown governor to a front-runner in 2004, Joseph Trippi, sounded almost enthusiastic. “Given his resources, it’s all sitting there for him,” Mr. Trippi said. “People are so sick of the polarization of politics that he could make the case that it’s time to move beyond the two parties and that he’s the one to lead us.”

Republican pollster Frank Luntz said his most recent surveys show that in a match up against Senator Clinton and a non-McCain, non-Giuliani Republican, Mr. Bloomberg polls in the mid-20% range.

Still, talk of a Bloomberg bid stirs laughter in some Republican circles in Washington.

“Is there a single American clamoring for a Bloomberg presidency? A single one?” a Bush campaign veteran, who is currently unaffiliated with any of the 2008 contenders, asks. “If he spends $500 million, they will piss it away. I’m not even sure if he gets 10%. And I have no idea who he hurts. … It’s not like Perot, who was a Texan and instinctively conservative.”

With Mr. Bloomberg dubbed “the sane Perot,” it is worth recalling that one party was significantly better prepared to respond to the Texas billionaire in 1992.

The deputy campaign manager for President George H.W. Bush that year, Mary Matalin, articulated official Washington’s obliviousness to the Perot factor in “All’s Fair,” her 1994 autobiography with her husband James Carville, who was a winning campaign manager for President Clinton.

“If we were slow on the uptake about Pat Buchanan, we were no quicker to respond to the whole idea of Ross Perot,” Ms. Matalin wrote. “He just wasn’t a factor in our lives. No one took him seriously.… Our first reaction, full of conventional thinking was, ‘This guy is incredibly rich, the people will resent that he is buying the election.’ … In fact, they had the opposite reaction; they granted him credibility because he wasn’t beholden to any special interests. So we misjudged his appeal from the very beginning.”

In 1992, while the Bush campaign’s official position was that they saw Mr. Perot as a non-factor and would ignore him, prominent Republicans including Vice President Quayle’s wife, Marilyn, White House Press Secretary Marlin Fitzwater, and Republican National Committee Chairman Richard Bond attacked him early. By contrast, in early 1992, Susan Thomases, a friend of the Clintons, was warning Mr. Carville that Mr. Perot “was hitting a sore nerve and could be a major factor in the race.”

Ordinarily, the two parties could rest easy, noting that even a two-term mayor might have trouble qualifying for the ballot in every state. But Mr. Bloomberg’s task may be eased by a grassroots organization pledged to support a centrist presidential bid, Unity08. One Unity08 organizer, Republican consultant Doug Bailey, is urging Mr. Bloomberg to run.

Mr. Bloomberg’s unusual combination of strengths — two electoral victories, a solid record as mayor, and tons of money — may be enough to overcome the huge obstacles to the success of a third-party bid.

All of this leaves Washington confused about whether to take the mayor seriously.