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Co-op City in the Bronx, with more than 15,000 apartments, is one of the places in New York City that has not yet been equipped with Verizon FiOS high-speed Internet.Credit Chang W. Lee/The New York Times

When New York City was preparing to let Verizon offer its high-speed FiOS Internet service in 2008, a city councilwoman, Gale A. Brewer, warned that “New Yorkers need to know whether they are going to wait three years or six years.”

For many frustrated residents, like Stephanie Brooks, it has already been seven years and there is still no FiOS in sight. Ms. Brooks, who owns a brownstone in the Bedford-Stuyvesant neighborhood of Brooklyn, says she doubts that she and her neighbors will ever have the kind of speedy Internet options available in other parts of the city.

“Why aren’t we getting the opportunity to get service and have a choice?” said Ms. Brooks, who pointed to more affluent neighborhoods that Verizon wired years ago. “They built around us.”

Ms. Brooks’s complaint goes to the core of an increasingly bitter dispute between Mayor Bill de Blasio and Verizon, one of the nation’s biggest and most profitable telecommunications companies, that may end up in a courtroom.

Mr. de Blasio has made universal access to affordable broadband a top priority for his administration to help bridge the so-called digital divide between those who can get online easily and inexpensively and those who cannot, as well as to make the city a competitive technology hub.

Verizon had agreed to have fiber-optic cable for FiOS pass all three million homes in the city by the end of last year. Lawyers for each side, however, are arguing about the definition of “pass.” The company says it has met the deadline. The city’s response: not even close.

FiOS remains unavailable in large swaths of the city, including the vast Co-op City complex in the Bronx, which comprises more than 15,000 apartments. “We’ve had some rather impassioned meetings with some of our residents who say, ‘We want Verizon,’ ” said Jeffrey Buss, general counsel for Riverbay Corporation, which manages Co-op City.

The city’s Department of Information Technology and Telecommunications released a scathing audit report in June concluding that Verizon “systematically refused to accept orders for residential service.” By the company’s admission, nearly one-fourth of the blocks in the city have no buildings wired for FiOS, the report said.

“We want them to make it available to everyone in every ZIP code and on every block so that everyone can get online, to do research, to do their homework,” said Maya Wiley, the chief lawyer for Mr. de Blasio. “We need our residents to get online.”

Ms. Wiley said that her staff was negotiating with Verizon and that the city would prefer not to sue the company for failing to fulfill its obligation. But, she added, “if that’s what we have to do, then that’s what we’ll do.”

Customers are still being told by Verizon that FiOS is “not available” to them, even after the company claimed to have wired the entire city. Ms. Brooks said that the last time she heard from Verizon was a letter in January telling her that the company was trying to figure out how to connect to her home.

One of Ms. Brooks’s neighbors, Barbara Cooke-Johnson, said that two years ago she escorted a Verizon representative door to door on her block, seeking permission to run fiber through the properties.

Ms. Cooke-Johnson, 75, a retired lieutenant for the city’s Emergency Medical Service, said she got about 10 neighbors to agree and thought that she would have upgraded from her poky DSL service by now. But she said that she never heard from Verizon again and remained stuck with an Internet connection so slow that it takes nearly 15 minutes to download her email.

Her 30-year-old grandson, Laquan, was particularly annoyed, she said, because “he’s on the Internet all the time.” Some neighbors complain that it is impossible for them to work from home without a fast, reliable connection to the Internet, Ms. Cooke-Johnson said.

Like most New Yorkers, they have been bombarded with ads for the FiOS service on television and, when they can get online, the Internet. Many of them have been persuaded, through those ads or personal experience, that FiOS is superior to competing services offered by the cable companies that had monopolies in different parts of the city before Verizon was granted a franchise to compete with them.

That franchise, granted in 2008 by the administration of Mr. de Blasio’s predecessor, Michael R. Bloomberg, was intended to promote competition that would improve service and hold down prices. One of Mr. Bloomberg’s deputies at the time, Robert C. Lieber, called it a “seminal moment in the cable industry in New York City that was long overdue.” He said that it would bring all of the city’s households “not only a true choice of providers and competitive prices, but also a host of benefits.”

But Common Cause, a public interest group, has criticized the contract as being too favorable to Verizon. “The problem is that there really weren’t significant penalties,” said Susan Lerner, executive director of Common Cause New York. That omission has left the city with two options, she said: Either “pressure Verizon to live up to its promises or sue.”

Today, many New Yorkers say what they most want is an answer from Verizon about when the fiber-optic cable will actually reach them.

After not hearing from Verizon since the letter in January, Ms. Brooks said she feared that the company had decided against upgrading her neighborhood.

For its part, Verizon says it has made no such choices.

The company said in a rebuttal to the city’s audit that it had met the requirement of having fiber in place throughout the city so that it could provide FiOS within a year to any resident who requested it. In virtually every case of a customer’s having waited more than a year, Verizon ran into an obstacle, such as an uncooperative landlord, the company said.

“I hope we’re not headed to litigation” with the city, said Kevin Service, senior vice president for network operations for Verizon. “We’ve invested an enormous amount of money” in building out the FiOS network, he said. “It’s absolutely in our interest to get it to as many subscribers as possible.”

Mr. Service said the company had completed the first phase of the project, laying the main lines, and was still “laying a lot of fiber” as part of the second phase, which involves getting wires through backyards and into rowhouses and high-rises.

As an example, Mr. Service cited the work necessary on a single block in Manhattan — 118th Street in East Harlem. “To get to the 10th floor in the middle of the block,” he said, “we’ve got to talk to not only that building, but the three buildings on one side and the four buildings on the other side.”

Mr. Service said he did not “see an end in sight” to that work, adding that it would probably not be completed during the term of the franchise, which runs until 2020. He said he expected Verizon would still be hooking up new customers “at least until then.” He said the company had “the pedal to the metal” with “no intention of slowing down.”

But several Verizon employees disputed the company’s assertions, saying that after wiring all of Staten Island, Verizon significantly slowed the expansion to the rest of the city. The company, they said, had been reducing the work force on the wire-line side of its business and had shifted resources to the fast-growing wireless phone business.

The union workers, members of the Communications Workers of America, spoke on the condition of anonymity because they are working without a contract amid contentious negotiations with Verizon. The union represents about 39,000 Verizon workers, down from about 45,000 when the workers went on strike in 2012 to protest the company’s contract demands.

Bob Master, an official of the union, said that the experiences related by the workers seemed to add up to “a pattern and a plan not to finish” the expansion.

City officials tend to agree. Scott M. Stringer, the city comptroller, said he believed that Verizon was “in violation of the intent of this agreement.”

“We as a city have to hold Verizon to the contract that they agreed on,” Mr. Stringer said. “This is the equivalent of making sure that every apartment had electricity in a bygone era. Access to the Internet is no longer a luxury; it’s a necessity of life.”