Several analysts argued that a three-way race with Mr. Bloomberg running as a third-party candidate could ultimately prove more of a threat to Democrats than to Republicans. Until he ran for mayor in 2001, Mr. Bloomberg was a lifelong Democrat, and his success in New York reflected his ability to draw Democratic votes: he is for abortion rights, gay rights and gun control.

Mr. Bloomberg’s aides said he has plans to travel to Missouri and to Florida, crucial swing states, in the near future.

On Friday, he filed papers with the city Board of Elections to change his affiliation, as the public focus on his future intensified. In the space of a few days, he appeared on the cover of Time and in a BusinessWeek special report, and he told an audience of Google employees that the country is “really in trouble.” He stood with Gov. Arnold Schwarzenegger of California as the governor told a crowd of reporters on Tuesday that Mr. Bloomberg “would be a great candidate,” in an appearance at Ceasefire, a conference on bridging the partisan divide, organized by the Annenberg School for Communication at the University of Southern California. “It’s all about fixing problems and creating a great vision for the future,” he said.

While Mr. Schwarzenegger has not been as critical as Mr. Bloomberg about the Republican party, he has differed with many in the G.O.P., and the two men have become increasingly linked. The governor supports stem-cell research and wants to extend health care to illegal immigrants in California. He supports a timetable for troop withdrawal in Iraq and has frequently derided Washington for failures on border control and other issues.

Adam Mendelsohn, the governor’s communications director, described their relationship as “ symbiotic."

At their joint appearance, the mayor was the big draw, as a throng of reporters and cameramen crowded around, peppering him with questions about whether he would seek the presidency.

Mr. Bloomberg, who has called it flattering to be viewed as presidential material, clearly enjoyed the attention. He joked and parried with the press, and when asked whether he and the governor might team up on a campaign ticket, suggested that the governor would challenge him to an arm wrestling contest to see which of them should run at the top, while he would consult the Constitution. (The Constitution says that the president must be “a natural-born citizen,” which is generally regarded as barring naturalized citizens.)