Caroline Kennedy, the daughter of slain United States president John F Kennedy, is pursuing the Senate seat of Hillary Clinton, who has been nominated as secretary of state.
Civil rights activist Al Sharpton said he had received a phone call from Ms Kennedy in which she expressed her interest in the seat.
Caroline Kennedy, 51, spent part of her childhood in the White House until her father was assassinated in 1963 and since then has kept a relatively low public profile.
She endorsed Barack Obama during his primary campaign against Hillary Clinton for the Democratic Party nomination and later helped the search for his vice-presidential candidate.
Mr Sharpton said he told Ms Kennedy he had not had any meaningful discussions with New York state governor David Paterson, who has the authority to appoint any replacement for Mrs Clinton, about her or any other potential candidate.
"I unequivocally disagree with those that say she is not qualified and could not bring needed leadership to this state and country," he said.
Mr Paterson told a news conference in the state capital Albany on Monday that he would not talk about any individual candidates.
Ms Kennedy's family is an American political dynasty. Her uncle, Robert F Kennedy, was a New York senator and a presidential candidate when he was assassinated in 1968.
Another uncle, senator Ted Kennedy of Massachusetts, is one of the most powerful members of the Senate.
Mrs Clinton is not expected to resign as New York's junior senator until the Senate considers her confirmation as the nation's top diplomat after Mr Obama takes office on 20 January.