A spokesman for Schumer did not provide comment for the story, although the report does cite "associates" close to the senator who indicate he is not interested in the job and has passed up other opportunities to run for executive offices in the past.

Schumer, 59, was first elected to the Senate in 1998. He is thought to aspire to become Senate majority leader if and when Sen. Harry ReidHarry ReidThis obscure Senate rule could let VP Mike Pence fully repeal ObamaCare once and for all Sharron Angle to challenge GOP rep in Nevada Fox's Watters asks Trump whom he would fire: Baldwin, Schumer or Zucker MORE (D-Nev.) steps away from the leadership position, although he will likely face a challenge from Majority Whip Dick DurbinDick DurbinThe Hill’s Whip List: Where Dems stand on Trump’s Supreme Court nominee Gorsuch rewrites playbook for confirmation hearings Gorsuch: I'm 'sorry' for ruling against autistic student MORE (D-Ill.).

Bloomberg's overtures to Schumer and other candidates apparently come despite public signs that Bloomberg favors New York City Council Speaker Christine Quinn as his successor. As news that Bloomberg had approached Clinton leaked last month, Bloomberg sought to reaffirm his support for Quinn at a public event.

“This woman has made an enormous difference in this city,” he said, according to The Wall Street Journal. “She’s a leader, and I have nothing but respect for her.”

During an appearance on MSNBC Tuesday, Bloomberg discounted the report and again credited Quinn.

"The article was so erroneous and it goes after one of the people who's really made a difference in this city, Chris Quinn, our Speaker of the City Council, who really has done a great job. Without her it really would have been a lot tougher, let me tell you," he said.

This post was updated at 9:28 a.m.