The efforts include both conventional advertising on television, radio and online along with a novel social media strategy. He has hired some of the biggest meme-makers on the internet, who are posting what look like real direct messages from Mr. Bloomberg (they are not) asking them to help “make me look cool.”

With African-American voters specifically, his campaign has highlighted parts of his biography that are likely to resonate, like his relationship with Mr. Obama. Campaign ads feature Mr. Bloomberg’s work to expand My Brother’s Keeper, an initiative by the Obama administration supporting boys and young men of color, and his efforts to reduce gun violence by requiring more scrutiny for arms purchases. The campaign also emphasizes his apology for the stop-and-frisk program.

Mr. Bloomberg’s events across the South last week drew large crowds — more than 1,000 people were in attendance at separate events in Nashville and Chattanooga, Tenn. — and often included large numbers of African-Americans. In interviews, many voters were aware that the issue of his record on racial discrimination was in the news. But they also cited the former mayor’s work with Mr. Obama and his gun control campaigns. And while his record on policing was a sore spot, some gave him credit for apologizing.

“He came out and apologized and said that’s not the kind of policy I would support as president,” said Sheree Johnson, 35, an educator who attended Mr. Bloomberg’s event in Houston. “The Christian in me says to forgive him for that. He acknowledged it. He was wrong.”

Dwight Smith, who works with the N.A.A.C.P. in Chattanooga and attended Mr. Bloomberg’s rally there, said he believed many black voters were focused more on the bigger-picture goal of beating President Trump than they were on blemishes in any candidate’s past. “Everybody makes mistakes,” Mr. Smith said. “And if you look at the mistakes Donald Trump has made versus the mistakes Mike Bloomberg has made, I think people are willing to let it go.”

Not everyone, though. Benjamin Dixon, an African-American podcaster, has shared on Twitter Mr. Bloomberg’s previous comments supporting stop-and-frisk with the hashtag #BloombergIsARacist. That sort of criticism is indicative of the kind of resistance Mr. Bloomberg will most likely continue to encounter.

However, Mr. Belcher said the sentiments of the activist community and the political punditry were often not a good proxy for how a majority of black voters consider matters like these. “Lord knows if African-Americans had absolute purity tests for people who had problematic issues in the past, we would never advance as people,” he said.