For decades, people in search of the best food in Rome have found their way into the arms of the Roscioli family. Atop a network of wine cellars between the ancient Jewish quarter and the Campo de’ Fiori flower market, descendants of Marco and Franco Roscioli run four businesses, each outstanding in its field: a bread bakery, a pastry shop, a wine bar and a salumeria moonlighting as a restaurant that has become one of most sought-after tables in the city.

Twice a day, the staff of Salumeria Roscioli sets up the tables that transform the deli into an informal but ambitious restaurant. There, they serve dishes based on an enormous pantry that includes semi-dried cherry tomatoes from Campania, culatello from Emilia-Romagna and fig-walnut bread made in the 200-year-old oven at their bakery around the corner, L’Antico Forno Roscioli.

The Rosciolis’ reach extends beyond Italy. You can sip a cocktail made with gin from India or Japan and twirl pasta tossed in vanilla butter and Spanish anchovies; the Caffè next door serves club sandwiches.

“Even having French wine and Spanish ham is unheard-of” at Rome’s classic trattorias, said Frank Falcinelli, a Brooklyn restaurateur. “Romans will die on that hill.”

But by pushing the parameters of tradition, as the brothers Alessandro and Pierluigi Roscioli have done for decades, the Roscioli brand has flourished. Pierluigi, 49, Alessandro, 53, and their sister, Maria Elena, 29, who has recently joined the family business, have cultivated an enviable collection of Italian ingredients, nurtured connections with chefs around the world and built a strong social media presence.

On their watch, Roscioli will open its first outpost, in SoHo, next month. Instead of expanding to Florence or Milan, the logical next step, the Rosciolis will try to recreate the quality, patina and — most important — pastas of their Roman empire in Manhattan. In the spirit of the original, this Roscioli will be a deli and wine bar on the ground floor and a restaurant in the basement, all tucked into a 19th-century brick townhouse in the old Italian enclave once called the South Village.

When it comes to the holy Roman quartet of pastas — gricia, cacio e pepe, amatriciana and carbonara — each paired with a different shape of pasta, “they never wander off,” said the chef Nancy Silverton of the Mozza group in Los Angeles, referring to the Rosciolis’ focus on consistency, quality and technique.

Roscioli spurred countless Italian food trends in the United States long before opening here, such as Americans’ fixations with burrata, cacio e pepe and maritozzi, whipped-cream-filled buns that have been made in Rome for centuries, but became a social media star after Roscioli introduced theirs in 2016.

“Roscioli is the Kevin Bacon of restaurants,” said Mr. Falcinelli, who cites it as a business and culinary model. “Everybody has a connection.”

The Roscioli maritozzo is one of the models for Bilena Settepani’s recently perfected version at Settepani Bakery in Williamsburg, Brooklyn. By coincidence, Ms. Settepani was at Roscioli Caffè in Rome, “getting one last maritozzo before my flight home,” when reached by this reporter.

In Italy, food and wine producers aspire to join Roscioli’s ever-changing collection of hundreds of cured meats, aged and fresh cheeses, and 2,500 wines.

“Getting into Roscioli is the dream,” said Stefano Papetti, an owner of De Fermo, a small biodynamic winemaker in Abruzzo. Being on the wine list there, he said, generated demand for his wines far beyond Italy.

But in New York, the challenge will be replicating the quality of ingredients that Roscioli is known for, not only in its pastas but also on its plates of exceptionally fresh cheeses like burrata, buffalo mozzarella and buffalo ricotta, and cured meats like mortadella and prosciutto — few of which can be served in the United States.

Fresh cheeses made from raw milk can’t be imported at all, though aged cheeses like Pecorino Romano can be. “The winds keep changing,” said Zach Allen, who has overseen food imports at the U.S. locations of Eataly and the fast-growing Florentine sandwich chain All’Antico Vinaio.

“The imports used to be so heat-treated and over-cured that they didn’t taste anything like the original,” he said, referring to regulations imposed by the U.S. Department of Agriculture on cured meats

Over decades, Mr. Allen has painstakingly developed relationships and recipes with U.S. producers who make what he needs, like a fatty, fennel-spiked Tuscan salami called sbriciolona for La Favolosa, a top-selling sandwich at the New York locations of All’Antico. “The Rosciolis are at the very beginning of that process.”

Alessandro Pepe, the head sommelier at Rimessa Roscioli, the family’s wine bar, is leading the New York opening, and said that while they are building relationships, the core ingredients for their classic pastas will be coming from Rome: aged pecorino and Parmigiano-Reggiano, their own dried and jarred tomatoes, olive oil and dry pasta. “They are the kind of dishes anyone could have on the shelves and make at the last minute,” he said.

As the Rosciolis have influenced Italian food in America, American food has influenced them from the beginning.

The eldest of eleven brothers from a family of shepherds in Le Marche, Franco Roscioli made his way to Kenosha, Wis., in 1953. When he returned a few years later, it was with two important innovations: margarine, and an industrial bread slicer.

He opened a bakery on the outskirts of Rome, specializing in American-style sliced sandwich bread. As the business grew, Franco brought his brothers to work in the capital. Marco — Alessandro, Pierluigi and Maria Elena’s father — moved to Rome and started baking in 1958, when he was 12 years old.

At the time, commercial enriched breads were made with cheap olive oil and pork fat, neither of them ideal substitutes for butter. Margarine, hydrogenated corn oil, came much closer.

It was also kosher. Marco Roscioli opened L’Antico Forno in 1972, and his challah, made with margarine, was a hit among the nearby Jewish community. “Their patronage was the economic engine that allowed the business to get its legs, survive, and keep it afloat in its first years,” Pierluigi Roscioli said.

The idea of turning the salumeria into a restaurant was born in the United States, he said, at the corner of Prince Street and Broadway in New York, where Pierluigi visited the food emporium Dean & DeLuca in its influential heyday. It not only stocked premium ingredients from around the world, but it also offered ready-to-eat food to enjoy at tables set up among the shelves.

It took 10 years of discussion to decide on an equivalent Roman menu. They couldn’t make soup and salad from the ingredients on hand, he said. “But we could do a really beautiful pasta.”

New York, he said is the only place outside Italy that he can imagine being a Roscioli. “It’s absolutely captivating,” he said. “I wouldn’t sacrifice the beauty of Rome for anywhere else but New York City.”