When Zohran Mamdani catapulted to a stunning victory in New York City’s Democratic mayoral primary, the triumph seemed a coming-of-age moment for the Democratic Socialists of America.

The group formed the backbone of Mr. Mamdani’s canvassing operation and played an essential role in pushing the nation’s largest city to embrace an unwavering progressive campaign agenda.

But for Mr. Mamdani to get elected in November, he may need to win over segments of the city’s business class, or at least persuade them that he intends no harm. Some of that effort has already been on display, creating some discomfort among his core supporters on the left.

Last week, Mr. Mamdani, a state assemblyman and a democratic socialist, met separately with skeptical members of the Partnership for New York City and with Black business executives, who grilled him over his socialist economic agenda and challenged him over some of his stances opposing the wealthy and supporting Palestinian causes.

Billionaires shouldn’t exist? Mr. Mamdani walked that back. A rent freeze for stabilized units? Yes, but it was a policy he might revisit after four years. His refusal to repudiate the term “globalize the intifada?” That, too, came under some revision.

For now, D.S.A. leaders and others on the left say that Mr. Mamdani has earned their trust and deserves a fair amount of latitude. They recognize that the best way to push their agenda is to have a powerful emissary like Mr. Mamdani leading the nation’s largest city.