The New York Post also reported that the daughter of slain President John F. Kennedy withdrew from consideration, citing unnamed sources.
The New York governor's office made no immediate comment and a spokesman for Kennedy could not be reached.
Kennedy, 51, spent part of her youth in the White House but has never held public office.
After President Barack Obama tapped Clinton to be his secretary of state, Kennedy launched an unusual public campaign to replace her. The move was seen as a possible continuation of the Kennedy political dynasty.
Clinton won Senate approval on Wednesday and was sworn in as secretary of state , a day after Obama became president. Clinton resigned from the Senate immediately after her confirmation.
The Times said Paterson was expected to announce his Senate choice on Saturday.
Paterson had said he was impressed with Kennedy but said he was also considering other candidates who were not as well known.
Her withdrawal would leave New York Attorney General Andrew Cuomo as a front-runner for the job. Cuomo is a former cabinet secretary under President Bill Clinton and the son of former New York Gov. Mario Cuomo.
Another possible candidate, Rep. Carolyn Maloney of New York City, was named to chair the Joint Economic Committee, possibly signaling she was out of the running.
The Times reported that Kennedy's decision to withdraw was prompted by concerns about the health of her uncle, Sen. Edward Kennedy, who was hospitalized after a seizure during an inaugural lunch for Obama on Tuesday.
She had spent much of her life trying to avoid the limelight until she endorsed Obama for president.
In an editorial published in The Times in January 2008, Kennedy wrote that Obama inspired her the way the way people had been inspired by her father and she also campaigned for him in his primary battle against Clinton.
Many Americans remember her as a lovable toddler in America's first family, of which she is the only survivor.
Her mother Jacqueline Kennedy Onassis died of cancer in 1994. Her brother John F. Kennedy Jr. died, along with his wife and sister-in-law, as he piloted a small plane to a family wedding at the Kennedy compound in Massachusetts in 1999.