As recently as 2014, nearly half the City Council accused the Walton Family Foundation, which is funded by the company’s founding family, of trying to curry favor with New Yorkers by donating “toxic money” to local efforts to support charter schools.

This time around, many city leaders either declined interview requests or did not return requests for comment on Jet’s growth plans in New York. Some are skeptical, even if they are taking a wait-and-see attitude.

“Walmart may think they have found a new, under-the-radar path into New York City by buying up businesses already here, but we should not be fooled,” Comptroller Scott M. Stringer, a Democrat, said in a statement. “We intend to be watching very carefully.”

Liberals hold at least as much sway over New York politics as they did when Walmart tried to open a store in Brooklyn in 2011. But since then, some activists say, their concerns about labor and the broader economy have moved beyond Walmart.

Maritza Silva-Farrell, executive director of the labor and community group Align, worked to sink Walmart’s plans to open a store in the East New York section of Brooklyn. She and others argued that Walmart’s low prices would endanger local businesses and that its lower wages would depress pay at other retailers.

Ms. Silva-Farrell said she was still concerned about Walmart’s business model. But Amazon, she said, is becoming “a world threat” as the company increasingly dictates the global supply chain and broadens into manufacturing.