New York City F.C.’s circuitous search for a permanent home — a seven-year quest that generated frustration for the team, exasperation among its supporters and endless ridicule from rival fans in Major League Soccer — has come full circle.

The team’s owners, in conjunction with a group of local developers, are nearing an agreement with New York City that would allow the team to construct a privately financed, 25,000-seat stadium in the South Bronx as part of a development project costing more than $1 billion.

The new stadium, the anchor of a plan that will replace parking lots and an elevator parts factory with affordable housing units, a new school, a hotel and retail stores, would rise just a short walk’s distance down River Avenue from the team’s current home at Yankee Stadium.

The group of developers, which includes Jorge Madruga’s Maddd Equities, says the project has the support of Mayor Bill de Blasio and James Patchett, the president of the city’s Economic Development Corporation. The E.D.C. said in a statement that it sees value in the proposal but cautioned that no formal agreement has been finalized.