Caroline Kennedy, the press-shy daughter of President John F. Kennedy, has decided to seek appointment to the Senate seat being vacated by Hillary Rodham Clinton, according to a Democratic source familiar with the decision.

Kennedy’s decision comes several weeks after her name was first floated as a possible replacement for Clinton, following a phone conversation she had with New York Gov. David Paterson. Paterson, who took over for Gov. Elliot Spitzer earlier this year, has the sole power to appoint the next senator.

In the intervening weeks there has been a behind-the scenes battle — that has occasionally spilled out publicly — between backers of Kennedy and those of Clinton.

The Clintonites have made clear that they remain less than pleased that Caroline Kennedy endorsed President-elect Barack Obama during the Democratic presidential primaries and have voiced concern that Clinton’s Senate seat would be handed off to a New Yorker who didn’t support her national bid.

The Kennedy forces — led by Kerry Kennedy, who, interestingly, is divorced from New York Attorney General Andrew Cuomo, the other frontrunner fot the appointment — have taken to the television airwaves to make the case for Caroline.

That case in a nutshell: she is the heir to the most famous political legacy in American politics, she has universal name identification and the ability to raise untold amounts of money, and she is personally close to the president-elect.

It was not immediately clear what prompted Kennedy’s interest in the race although her active support of Obama — she endorsed him and then helped lead the search for his vice president — suggests that the election of the Illinois Senator may have played a significant role.

Kennedy’s decision to make her interest known marks a significant break with her past shunning of the public eye. While Kennedy is almost certainly now the frontrunner for the appointment, her acknowledgment of her interest now transforms her from a largely private to an almost-entirely public figure.

Should she win the appointment, Kennedy will face a 2010 special election for the remaining two years in Clinton’s term and then another election in 2012 for the full six year term.

Republicans insist that despite New York’s decided Democratic lean they will make a serious run at the seat in 2010 with former New York City Mayor Rudy Giuliani and Rep. Peter King the most well-known potential candidates.

Aside from Kennedy and Cuomo, Democrats mentioned for the appointment include: Reps. Kirstin Gillibrand, Carolyn Maloney, Brian Higgins and Steve Israel as well as Buffalo Mayor Byron Brown and Nassau County Executive Tom Suozzi.

The news of Kennedy’s interest was first reported by Nicholas Confessore of the New York Times.

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