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. …

During their conversation, Mrs. Clinton left little room for doubt: she was not interested in seeking the mayoralty, people briefed on the call said.