Mitt Romney on Tuesday had his first meeting with Michael Bloomberg, the independent New York City mayor whose support is being courted by those on both sides of the aisle in the 2012 White House race.

New York City Mayor Michael Bloomberg (AP Photo/Karly Domb Sadof)

News of the meeting was first reported by Michael Barbaro of the New York Times

Both Romney and President Obama have reached out to Bloomberg in recent weeks.

Sen. John McCain (R-Ariz.), a top Romney backer, paid a visit to New York City Hall late last month, the New York Post reported:

“I just came in to pay my respects to the mayor. He and I are old friends from many years back,” McCain told the Post as he left City Hall. “I told him that I just spent last weekend with Romney and I thought that Romney was on message . . . and tried to convince the mayor that we’ve got a winning campaign.”

Obama, meanwhile, hosted what Bloomberg described as “a very pleasant lunch” for the three-term mayor at the White House in mid-March.

While an endorsement from Bloomberg could help either Obama or Romney appeal to independents in the fall, both candidates have their differences with the mayor on key issues.

Bloomberg is a staunch gun-control advocate, while Romney pledged at a National Rifle Association conference last month that he would “defend the [gun] rights President Obama ignores or minimizes.”

And Bloomberg has criticized as “theatrics” Obama’s proposed Buffett Rule of taxing millionaires at a rate of at least 30 percent, a major proposal in Obama’s reelection bid.