According to the New York Times, Bloomberg, New York City's mayor since 2002, admires Clinton for her ability to work with President Obama, a former rival, and wanted a politician with similar star quality and experience to succeed him.
The NY Times writes:
In a phone call confirmed by three people, Mr. Bloomberg encouraged Mrs. Clinton to consider entering the 2013 mayor’s race, trading international diplomacy for municipal management on the grandest scale. She would, he suggested, be a perfect fit.
Much about the call, which occurred some months ago, remains shrouded in mystery. But Mr. Bloomberg’s overture to the former first lady highlights the level of his anxiety about the current crop of candidates, his eagerness to recruit a replacement who can rival his stature and his determination to become a kingmaker in the political arena he will soon exit.
Bloomberg's interest in Clinton will undoubtedly make things a bit awkward for New York City Council Speaker Christine Quinn, the woman he is widely expected to publicly endorse.