Caroline Kennedy, a daughter of America’s most storied political family who for many years fiercely guarded her privacy, is considering whether to pursue the Senate seat expected to be vacated by Hillary Rodham Clinton early next year, a family member said Friday.

“I believe that she is considering it,” said her cousin Robert F. Kennedy Jr., who has spoken to Ms. Kennedy about the matter during the past week. “A lot of people the last couple of weeks have urged her to do it.”

Ms. Kennedy called Gov. David A. Paterson on Wednesday to discuss the position, Mr. Paterson confirmed Friday. The governor will choose a replacement for Mrs. Clinton upon her expected confirmation as secretary of state next month.

“The conversation was informational,” Mr. Paterson said. “She did not express an interest in the Senate, but we talked about the Senate, so I got that she was just trying to get some information to determine whether or not she would like to have an interest in it. And that was it.”