Ms. Kennedy made headlines around the world last week after alerting the governor that she wanted the job. She then began a public tour, meeting with political leaders around the state, and quickly cemented herself as the dominant contender for the seat.

Image Caroline Kennedy is actively seeking appointment to the Senate seat being vacated by Senator Hillary Rodham Clinton. Credit... Don Heupel/ Associated Press

“Precisely because there is no campaign or election, she should be more willing to disclose and subject herself to a greater level of public scrutiny than is required,” said Dick Dadey, executive director of Citizens Union, a nonpartisan watchdog group. He noted that other major contenders for the Senate seat  officeholders like the attorney general, Andrew M. Cuomo, and Representative Kirsten Gillibrand  have mounted runs for office and filed public disclosures before.

Others wonder if Ms. Kennedy’s unwillingness to disclose personal information suggests she lacks the stomach for the kind of intrusive questions that could come her way as a candidate in 2010.

“If this were an open primary, and all the people seeking that position had to run, she’d have to make all those disclosures, so why not in the appointment process?” said Bob Edgar, president of Common Cause, a watchdog group that lobbies for tighter ethics rules. “She can’t simply ride in on her name recognition or place in history. The voters and people of New York deserve that full disclosure.”

Fred Wertheimer, president of Democracy 21, another watchdog group, warned that requiring financial disclosure by “anyone who is speculated about” for such a vacancy could be untenable. “I would think it would be up to her,” he said. But he called Ms. Kennedy’s campaign for the appointment “kind of unique.”

So far, on her tour, Ms. Kennedy has taken just 11 questions from reporters, has granted no interviews, and responded only in writing to inquiries about her positions on significant issues.

“She needs to deepen the public’s idea of who she is,” said Paul Light, a professor at New York University’s Wagner School of Public Service. “To the extent she can be more transparent, she dispels the notion that it’s all about her name. We obviously know that she’s quite wealthy, but beyond that, we don’t know much about where she gets her income, how she’s invested, whether she has followed her own principles in her investing activities, and so forth. That would be very useful to know.”