Mayor Bill de Blasio has begun to tell people privately that he plans to run for governor of New York next year, according to three people with direct knowledge of his conversations with fellow Democrats and donors.

Mr. de Blasio, who has been a polarizing figure during his two terms in office, has also sounded out trusted former aides about their interest in working on a potential campaign, according to two people who are familiar with those contacts, and has made other overtures to labor leaders about a possible bid. His longtime pollster conducted a private survey to assess Mr. de Blasio’s appeal beyond New York City. And publicly, too, he has increasingly made it clear that he wants to remain in public life.

“There’s a number of things I want to keep working on in this city, in this state,” Mr. de Blasio said last week, noting his interest in public health, early childhood education and combating income inequality. “That is going to be what I focus on when this mission is over. So, I want to serve. I’m going to figure out the right way to serve and the right time to serve.”