The newspaper's source is a person close to New York Gov. David Paterson, who will be appointing Clinton's successor to the seat.
A source close to Kennedy, however, tells CBS News Director of Political Coverage Steve Chaggaris that Kennedy likely did not pull out over tax or housekeeper issues but over a more "serious" personal issue.
According to the source, Kennedy spoke to Paterson late yesterday afternoon to say she was removing herself from consideration. During the conversation, Paterson did not tell Kennedy she was not going to be appointed to the seat. In fact, the source said, Kennedy's sense was that the governor intended to appoint her.
Paterson gave Kennedy 24 hours to rethink her decision, the source said, and she subsequently wrestled with it. She decided to stand by her decision late last night.
The governor announced late this afternoon that he will make a Senate announcement tomorrow at noon. Among the leading contenders are New York Attorney General Andrew Cuomo and Congresswoman Kirsten Gillibrand.
Kennedy said yesterday she was removing herself from consideration for the seat due to a "personal situation." As CBS News reported previously, an aide said the issue was not related to the health problems of her uncle, Sen. Ted Kennedy.
According to the CBS News source, the reason Kennedy dropped out of contention truly is personal, and is something that only she and her immediate family are aware of.
There have been numerous conflicting explanations for Kennedy's decision, and the Times story includes comments from a Kennedy aide that suggest any tax or housekeeper issues may not have been significant enough to derail Kennedy's bid.
The aide suggested the issues had been acknowledged by both sides and "the Kennedy camp and the governor's operation had been discussing how to publicly disclose the issues" if she were selected.
The newspaper did not offer details on Kennedy's alleged tax and housekeeper problems.
Other storylines about the Kennedy withdrawal persist. CNN is reporting that Paterson belived Kennedy was not "ready for prime time" and had no intention of appointing her.
The CNN story cites a Kennedy ally saying that Paterson's disfavor was not a factor in her withdrawal. But the New York Post, which broke the Kennedy story yesterday, reported that she did know she wasn't going to be chosen and withdrew simply to save face.