Mr. Bloomberg, who declined to be interviewed for this article, said in a statement that as he considered running, he kept hearing criticism over the policy and his defense of it.

“The more I listened, the more I began to accept what I had struggled to admit to myself: they were right, and I was wrong,” he said in the statement. “I believe when you get something wrong, you stand up and admit it, and so I started working with my team on the speech.”

Still, the issue keeps coming up, and Mr. Bloomberg trails other candidates among black voters. He was asked about stop-and-frisk in recent interviews with Stephen Colbert and on “The View.” When Democratic-leaning black voters nationwide were asked which candidate they would “definitely not consider supporting,” 17 percent named Mr. Bloomberg, according to a Washington Post-Ipsos poll this month. Only one candidate, Representative Tulsi Gabbard of Hawaii, had a higher level of disapproval.

Several prominent black leaders in New York City said in interviews that they believed that Mr. Bloomberg’s apology was, as he has suggested, the result of a gradual shift. They said that they had, over time, tried to convince Mr. Bloomberg that the policy was far more harmful to young men than the mayor might have realized.

His supporters said that Mr. Bloomberg had viewed stop-and-frisk as part of his broader campaign to save lives, which included a ban on smoking in restaurants, a war on soda and salt, and attempts to get weapons off the streets through gun control measures, and that he trusted his police commissioner, Raymond W. Kelly.