The people spoke on the condition of anonymity Monday because neither Kennedy nor Paterson have acknowledged she is seeking the position.
If appointed by Paterson, the daughter of President John F. Kennedy would hold the seat once occupied by her late uncle, Robert F. Kennedy.
There was no immediate comment from the Kennedy family or from Paterson.
Paterson has sole authority to name a replacement for Clinton, who was first elected in 2000 and re-elected by a wide margin in 2006.
Over the past week, Kennedy has reached out to several prominent New York Democrats to tell them of her interest in the Senate seat. They included Joel Klein, chancellor of the New York City Department of Education; Kennedy worked closely with Klein as executive of Office of Strategic Partnerships for the New York City Department of Education, where she raised some $65 million for the city's schools.
Other Democrats who appear to be on Paterson's short list include New York Attorney General Andrew Cuomo, who won't say publicly if he's interested.
One of the early front-runners, Rep. Nydia Velazquez of Brooklyn, took herself out of the running Friday.
Paterson, a Democrat, will appoint someone to fill Clinton's seat for two years if she is confirmed as secretary of state. He is expected to tap someone who can raise a lot of money and help him politically when they run together on the 2010 ticket.
Other names in the mix include Nassau County District Executive Tom Suozzi, who is also a possible choice for lieutenant governor with Paterson in 2010; Buffalo Mayor Byron Brown; Bronx Borough President Adolfo Carrion Jr.; and Reps. Carolyn Maloney, Steve Israel, Jerrold Nadler, Kirsten Gillibrand and Brian Higgins.
Republicans wasted no time in criticizing Kennedy as unqualified for the job and unfamiliar with the state.
"If anything, it makes me more determined to run," said Rep. Peter King, a Long Island Republican who has already expressed his interest in the seat.
"As far as record of achievement I strongly believe that I'm much more qualified, much more experienced, and have an independent record," said King. "Nothing against Caroline Kennedy but I don't think anyone has a right to a seat."