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Asked recently whether he wants to be president, Mayor Michael Bloomberg responded in that brusque, smart-alecky way New Yorkers are famous for: "Which letter of the word 'No' do you not understand?"

But three months into his second term, the Republican billionaire who used to merely tolerate the glare of the national spotlight is thrusting himself onto a larger stage more frequently.

In the past month alone, without prompting, he has questioned the sincerity on Capitol Hill, warned against the erosion of First Amendment rights and renewed his push for tighter federal gun control. He has met with four members of President Bush's Cabinet and made two trips to Washington, one of which included a national policy speech about housing that touched on racial and economic inequality.

So does he want to be America's president, or is he trying to imitate his predecessor, Rudolph Giuliani, and become known as America's mayor?

Bloomberg's senior aides acknowledge his recent shift beyond city limits, but attribute it to the difference between his two terms. A political novice when elected in 2001, he spent his first years focusing on more local issues, such as repairing the city's tattered post-Sept. 11 economy and winning mayoral control of the beleagured school system.

"Now, New York really has become a model for urban America, and now there are issues at the federal level that the city needs addressed," says Kevin Sheekey, a deputy mayor and Bloomberg's top political strategist.

Aides say the mayor is also more comfortable now, and has the political capital and the confidence to use City Hall as a bully pulpit on issues that concern him.

Giuliani earned his wider recognition under tragic circumstances, the attack on the World Trade Center. With Bloomberg, it is coming more slowly and deliberately.

To combat gun violence, his administration plans to sue gun dealers in other states. His health department has created a diabetes registry to track patients with the disease. He believes mixed-income housing is vital to preserving American cities.

Gun control, public health and housing happen to be issues that could play well in a presidential campaign.

"The nation now knows Michael Bloomberg -- they've come to respect what he's done and he's become a symbol of New York," says Mitchell Moss, a public policy professor at New York University and occasional Bloomberg adviser. "Around the country, I get the question, `Why doesn't Michael Bloomberg run? He'd be a great president."'

Bloomberg, who built his $5.1 billion fortune from the financial information company he founded in the early 1980s, says he plans to leave politics for a life of philanthropy after his second term ends in 2009.

But speculation about higher office for the Democrat-turned-Republican refuses to die.

Such talk can enhance Bloomberg's effectiveness, in the short-term and the long-term.

"He can get great mileage from the speculation that the country considers him a candidate for president," says William Cunningham, a government operative who was Bloomberg's communications strategist until recently. "If along the way it boosts your profile, that's helpful too."

So which party would suit Bloomberg if he were to run for president?

He often disagrees with the GOP but does not seem willing to part with it completely, telling a group of Manhattan Republicans this month: "I couldn't be prouder to run on the Republican ticket and be a Republican."

But the moderate mayor supports abortion rights and gay marriage, and was a Democrat until he switched to run for mayor in 2001.

Analysts say he is far too liberal to appeal to Republicans nationally, so could he break away and run as an independent?

"People don't view him as a Republican or Democrat, and he has the resources to run without having to worry about funding," says Ed Rollins, the Republican strategist who ran billionaire Ross Perot's campaign in 1992.

"If we reach a point in a year from now that people are not satisfied with the two choices, and they continue to find this national disgust over partisan politics, a guy like Bloomberg could certainly step into the fray."

The 2008 field is already crowded with potential candidates from New York, including the mayor's fellow Republicans Giuliani and Gov. George Pataki, plus Democratic Sen. Hillary Rodham Clinton.

The mayor's aides and associates say they are taking him at his word that he is not interested in a White House bid. But just in case the mayor decides to change his answer to that question, Sheekey says coyly: "Keep asking."