Mayor Michael R. Bloomberg has long struggled to imagine a successor with the combination of star power, experience and grit to fill his shoes.

But not long ago, he was struck by an inspiration: Hillary Rodham Clinton, the retiring secretary of state.

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.