Once Clinton is confirmed to President-elect Barack Obama's cabinet, New York Gov. David Paterson will appoint someone to fill the seat for two years.
The Kennedy family's connections and history cannot force Paterson to choose Caroline, but the family's strong support could make it difficult for him not to.
Robert Kennedy said the family would come out en masse for her if she does get the appointment and has to run for election in 2010.
"If she runs, you will see more Kennedys than you have ever seen in your life," he said.
An environmental lawyer who took himself out of consideration for the Senate seat earlier this week, Robert Kennedy said he has spoken to his cousin about the position and is one of "many, many people" urging her to seek it.
He also offered a policy rationale for her in the role: education.
"She's probably one of the leading advocates in the nation on public education. She feels a lot of the issues she's worked on are in danger of being shunted aside because of the economic crisis," he said.
Democrats said Caroline Kennedy and Paterson have already spoken about the Senate seat, and she is interested.
After two New York Democrats said Kennedy and the governor are expected to meet privately to discuss the matter Saturday, the governor's spokesman said they do not have a meeting planned.
Kennedy is the daughter of President John F. Kennedy and a niece of brothers Edward and Robert. Robert Kennedy held the New York seat from 1965 until his assassination in 1968. Edward Kennedy has been a senator from Massachusetts since 1963.
As a prominent member of the Kennedy clan, Caroline Kennedy Schlossberg is the kind of high-profile, historic figure who could overshadow many other New York politicians hoping to be Paterson's choice.
The governor has said he is in no rush to make a decision, and Clinton is not giving up the seat before she is confirmed as President-elect Barack Obama's secretary of state.
"The governor has not yet reached out to any potential candidates," said Paterson's spokesman, Errol Cockfield. "He has been approached by several candidates. Any discussions related to that selection are private and the governor will not comment about speculation before a decision is made."
Whoever Paterson appoints would serve for two years and then have to run in a special election in 2010, along with Paterson and New York's senior senator, Charles Schumer. The candidate would then have to run again in 2012.
Kennedy has strong connections to incoming Obama administration officials - though Obama transition spokeswoman Stephanie Cutter insisted they were not involved in any way with the search for the next U.S. senator from New York.
As a prominent booster of Barack Obama's presidential bid, Kennedy spent much of 2008 taking bigger steps onto the public stage.
As famous as she is, she always has been viewed as almost painfully shy. She does not like talking about herself, nor does she appreciate those who do.
She met her husband, Edwin Schlossberg, while working at the Metropolitan Museum of Art. They married in 1986 and have three children.
She made a splash in early 2008 by writing an op-ed column for The New York Times declaring her support for Obama, saying he had the potential to be as inspirational to Americans as her father was in the 1960s. She also spoke at the Democratic National Convention.
She then hit the campaign trail with Obama, and worked on the vice-presidential search that eventually settled on Joe Biden.
Caroline Kennedy is easily the most famous contender for Clinton's Senate seat, but there are plenty of others. New York Attorney General Andrew Cuomo is widely known in the state. Paterson could also pick Buffalo Mayor Byron Brown or Tom Suozzi, a Long Island elected official.
There are also a number of House members in the running, including Reps. Carolyn Maloney, Kirsten Gillibrand, Steve Israel, Brian Higgins, Nydia Velazquez and Jerrold Nadler.