MSNBC also reported that a person close to Kennedy denied that she was out of the running.
Spokesmen for Caroline Kennedy and for Gov. David Paterson, who will make the appointment to fill the Clinton seat, wouldn't comment.
The hours of mixed signals were the latest twist in the Kennedy effort, which began with popular support that withered after she drew criticism in her brief tour and early press interviews. The reports were a shock to supporters and to Paterson, who had appeared likely to appoint her to the seat by Saturday.
Edward Kennedy, who has been treated for an aggressive brain tumor, suffered a seizure Tuesday at the inauguration of President Barack Obama.
The New York Post was first to report his niece's withdrawal from the Senate contest.
The New York Times cited a source it didn't identify as saying Kennedy withdrew out of concern for her uncle and his illness. But the Post cited an unidentified source as saying she dropped out because she learned Paterson had decided not to choose her.
The AP initially reported Kennedy had withdrawn from the race but corrected the story about an hour later after the person who gave that information said it was an error.
The Times and Post had no immediate comment on their stories late Wednesday.
Paterson said that on Wednesday he would begin reviewing the candidates' responses to the 28-page questionnaires he had asked applicants to complete. The forms asked about personal finances and other background issues, many of which Kennedy has long shielded from the public.
Kennedy, an author, lawyer and fundraiser for New York City schools, has long guarded her privacy, and the questionnaires were expected to include some closely guarded Kennedy financial data.
Paterson had said he thought the candidates' responses would be confidential because it was his personal request that they fill them out. But the state's open-government expert and good-government groups told the AP that once the forms were written and submitted to the governor at least some of the responses would be subject to public review under the state Freedom of Information Law.
Kennedy jumped to the top of statewide polls in early December, but her public support waned following a brief upstate tour and a few press interviews. She was criticized as reluctant to answer questions, and her knowledge of New York and its issues were suspect. She was also mocked nationwide for her frequent use of "you know" and "um" in interviews and was branded a lackluster campaigner.
Several other candidates are under consideration, including Attorney General Andrew Cuomo, who surpassed Kennedy in statewide polls last week.
Paterson said Cuomo had outstanding credentials for the job. Cuomo was the housing secretary under President Bill Clinton. Cuomo was elected attorney general in 2006 and has since led national reforms in the student loan industry and had a role in reining in corporate spending on Wall Street.
Cuomo is also the most popular elected politician in New York in polls -- higher than Paterson, whose approval rating, while still high, has been slipping.
Other contenders include Reps. Carolyn Maloney, of New York City, and Steve Israel, of Long Island, along with a strong upstate candidate, Rep. Kirsten Gillibrand, whose district runs along the Hudson Valley. Other hopefuls among the 10 or 20 Paterson said were under consideration include Reps. Jerrold Nadler and Brian Higgins and Buffalo Mayor Byron Brown.
The Kennedy reports came hours after Maloney, some Democrats' top choice, was named chair of the Joint Economic Committee in Congress. That's a significant move because Paterson had made it clear the next senator's top job should be to help land a federal stimulus package to help New York out of its historic fiscal crisis.