Another person who had advised Mr. Paterson said that Ms. Kennedy could offer political advantages to the governor, who was elevated to his position after Eliot Spitzer resigned in March and in two years must ask voters to actually elect him as governor.

Image Ms. Kennedy with Barack Obama and Senator Edward M. Kennedy in January. Credit... Damon Winter/The New York Times

“The upside of her candidacy is that the 2010 ballot will read Kennedy - Paterson,” said one of those advisers, who was granted anonymity to speak candidly about the governor’s thinking. “David craves national attention and money. If you connect the dots, it leads to her.”

For Ms. Kennedy, an appointment to the Senate would open a historic and exceedingly high-profile chapter to a life largely shielded from public view, and comes at a poignant time for her personally.

Her uncle, Senator Edward M. Kennedy, is struggling with terminal brain cancer, and his illness has forced members of his extended family to contemplate the possibility that the Senate could be left without a Kennedy for the first time in a half century. Mr. Kennedy has encouraged his niece, to whom he talks nearly every day, to pursue Mrs. Clinton’s seat, a spokesman for the senator, Anthony Coley, said. Associates of the senator say he has made it clear he would not pressure her to do so. Still, they said nothing would make him happier or prouder than having his niece in the Senate, which  far more than the White House  has been the core of the family’s long record of public service.

Other members of the family, especially her cousin, Robert F. Kennedy Jr., have also strongly encouraged Ms. Kennedy, who, if she were appointed, would become the first woman to lead the Kennedy dynasty, whose most successful and visible members have been men. Her brother, John F. Kennedy Jr., who died in a plane crash in 1999, had once been urged to run for the seat, which was held by their uncle, Robert F. Kennedy.

Ms. Kennedy, who initially seemed taken aback by questions about whether she would be interested in the position, has grown increasingly excited about and focused on the opportunity in recent days, those who have talked to her said. She has moved aggressively into campaignlike mode, albeit with careful attention to political protocol.

On Monday, she called dozens of political figures to let them know she was interested in the job. Besides Mr. Paterson and Christine C. Quinn, the New York City Council speaker, Ms. Kennedy called upstate officials like Representative Louise M. Slaughter and Byron Brown, the mayor of Buffalo; the Rev. Al Sharpton, the civil rights leader; and Charles E. Schumer, New York’s senior senator.