They court him over coffee and drop by when they are in town. They dangle invitations for golf and enlist friends to put in a good word.

In an election year when partisanship has burned white hot and the economy has sputtered, two presidential candidates who agree on very little, Mitt Romney and Barack Obama, have reached a rare consensus: They are both determined to score the endorsement of Mayor Michael R. Bloomberg of New York, whose name is all but synonymous with Wall Street clout and nonpartisan politics.

Now, their pursuit of the billionaire mayor is headed into overdrive, with both campaigns making the kind of conspicuous ring-kissing gestures that are reserved for their most sought-after political allies, even though the candidates publicly disagree with the mayor on a range of issues.

On Tuesday, Romney showed up at the mayor’s philanthropic foundation in Manhattan for a secret breakfast meeting. Over coffee and juice, Romney made clear that he was there to pick the mayoral brain: “Tell me what’s on your mind,” he told Bloomberg, according to aides briefed on the 30-minute discussion, which touched on immigration, gun control and education policy.

In its own public display of affection, the White House invited Bloomberg for a round of golf with two members of the Obama administration: He played with Vice President Joe Biden and his son Hunter at a private club in suburban Maryland on Friday and with Defense Secretary Leon E. Panetta at a military base in Northern Virginia on Saturday, according to people informed of the outings.

The aggressive wooing of Bloomberg, an independent who did not take sides in the last presidential race, highlights the outsize role that Wall Street, as both a symbol and purveyor of campaign contributions, is likely to play in the 2012 election.

For all of his achievements, Bloomberg is, first and foremost, a business mogul, whose deep knowledge of and close relationships within the world of high finance could carry significant weight at a time when many in industry feel under siege.

“If you want to have friends in the financial sector and you want to shape public opinion in the business capital of the world, then the guy you turn to, the imprimatur you want, is that of Michael R. Bloomberg,” said Hank Sheinkopf, a longtime New York political operative who is close to the mayor.

During their meeting, Bloomberg and Romney discussed Wall Street’s frustrations with Obama, who has chided bankers for their practices and called for tighter regulation of their industry, according to a person with knowledge of the conversation, who like many interviewed for this article, spoke on the condition of anonymity for fear of upsetting the high-powered leaders.

Bloomberg, who has lent his reputation for common-sense government and his prowess for fundraising to dozens of candidates from both parties over the past decade, feared that an endorsement in the 2008 race might have negative repercussions for the city he oversees.

But as his mayoral term winds down, he has told advisers that he is willing to back a candidate this time around, touching off an intense competition for his support in the general election.

“I’ll see down the road,” the mayor said, coyly, Tuesday when asked about an endorsement.

Describing his impressions of Romney and Obama, he made clear that he sees a wide gap between them.

“They’re very different, and they give the public a real choice,” he said. “It’s hard to argue that you can’t tell the difference, if you will. They run the spectrum on lots of different issues.”

Bloomberg appears to relish the attention from the campaigns, which had crammed his calendar with boldface names. There was a lunch with Obama at the White House in March, not to mention a golf game with the commander in chief not long ago.

Neither candidate, however, is relying on his own charms to win over Bloomberg. In late April, a high-profile supporter of Romney, Sen. John McCain of Arizona, talked up the presumptive Republican nominee to the mayor during a visit to City Hall, according to aides.

At the same time, aides to the president periodically reach out to the mayor’s office to take Bloomberg’s political temperature: Biden stopped by for coffee at Gracie Mansion, the mayor’s official residence, in 2010.

To commit to either candidate, Bloomberg will have to make peace with, or at least overlook, a raft of major policy disagreements, several of them on issues for which the mayor has deep-seated and passionately held views.

Romney, for instance, vigorously opposes legalized abortion and has called for the elimination of federal financing for Planned Parenthood. Bloomberg is an avid defender of those rights, and, this year alone, donated $250,000 to the Planned Parenthood when the organization came under fire from conservatives.

Obama pushed for the extension of the middle-class tax cuts created under President George W. Bush, which Bloomberg has called fiscally irresponsible. And Obama has used language to describe Wall Street that has offended the mayor, who has scolded the president’s political party for “pitting Americans against each other in a game of class warfare for their own political purposes.”

Bloomberg, never known for suppressing his opinions, is hardly shying from these disputes with the candidates. During his meeting with Romney, he acknowledged their policy differences, especially over the issue of gun control, according to people told of the discussion. Bloomberg has financed a coalition to push for stricter requirements on background checks for gun buyers andB the concealment of firearms. Romney has railed against the creation of rules that he said “burden lawful gun owners.”

Romney was careful neither to ask for Bloomberg’s support, nor to discuss the possibility of choosing Bloomberg as his running mate, a possibility regarded as highly unlikely.

“It was a first date,” said an adviser to one of the men.

And as with any debut, Romney was intent on leaving the best possible impression. When pressed about the meeting Tuesday, he was effusive in his praise for New York City’s revitalization and Bloomberg’s role in it.

“It’s quite an extraordinary story,” Romney said.