WASHINGTON - Caroline Kennedy - daughter of a slain president, niece of two U.S. senators, and zealous guardian of her own privacy - is interested in the U.S. Senate seat once held by one of her uncles, her cousin says.

Robert F. Kennedy Jr. said he and the rest of the powerful Kennedy clan are urging Caroline Kennedy to seek the New York governor's appointment to the Senate seat now held by Hillary Rodham Clinton - and added she is ready and seriously considering it.

"I know she's interested," Robert Kennedy, who himself was prominently mentioned for the seat, said in a telephone interview Friday. "She spent a lot of her life balancing public service with obligations to her family. Now her children are grown, and she is ready to move onto a bigger stage."

Once Clinton, in line to become secretary of state, 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 family's connections and history cannot force Paterson to choose Caroline Kennedy, but the family's strong support could increase pressure on him to pick her over lesser-known contenders. For Kennedy, seeking the Senate seat would be a significant departure from the life she has lived until now, protecting her family's privacy.

Robert Kennedy said his extended family would come out en masse for her if she does get the appointment and runs 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 last week, Robert Kennedy said he is one of "many, many people" urging her to seek it, partly because of her lifelong advocacy on education issues.

"She's probably one of the leading advocates in the nation on public education," he said. "She feels a lot of the issues she's worked on are in danger of being shunted aside because of the economic crisis."

Democrats said that Caroline Kennedy and Paterson have already spoken about the Senate seat, and she is interested.

Kennedy is the daughter of President John F. Kennedy and a niece of brothers Edward and Robert. Robert F. Kennedy held the New York seat from 1965 until his assassination in 1968. Edward Kennedy has been a senator from Massachusetts since 1963 and still uses the old Senate desk that Caroline Kennedy's father used as a senator from Massachusetts before he won the presidency in 1960.

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 to the cabinet post.

Caroline Kennedy made a splash in the presidential campaign early this year by declaring her support for Obama in an opinion piece she wrote for the New York Times. She said 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 his vice presidential search that settled on Delaware Sen. Joseph R. Biden Jr.