Life may soon be getting harder for New York City’s largest publicly traded company when it comes to dealing with one of its biggest customers—the city of New York.

The de Blasio administration is putting Verizon Communications on notice that business it used to do with city agencies will now go through City Hall, which could choose to block discretionary transactions if it deems the company a “bad actor.” The move is aimed at pressuring the telecommunications giant to make good on its overdue promise to deliver broadband service citywide.

In a meeting at the end of June, Mayor Bill de Blasio told commissioners and agency heads they must inform the Department of Information Technology and Telecommunications of all major contract negotiations with Verizon and other service providers. The mayor’s counsel, Maya Wiley, will issue final approval of any discretionary deal.

It’s no small threat: Verizon’s voice and data business for government agencies has totaled close to $650 million during the past five years, according to city estimates.

The edict reflects the administration’s frustration with Verizon. An audit last month found the company had not lived up to its 2008 franchise agreement to make its fiber-optic cable-television service available to all New Yorkers.

The de Blasio administration’s new protocol has been described by insiders as an attempt to keep Verizon from continuing with business as usual while failing to make good on its FiOS franchise commitments—or even acknowledging the shortcomings of its FiOS rollout.

Verizon, in a 30-page addendum to the audit report, disputed almost all its findings.

FiOS is of particular concern to the mayor because he has made universal broadband access a key component of his effort to reduce income inequality. The franchise agreement, reached during the Bloomberg administration, not only calls for the build-out of New York’s lone citywide fiber-optic network, but aims to spur competition by bringing choice to neighborhoods currently served by a single cable television and high-speed Internet provider.

City officials say they are not making threats. But they want Verizon to know the game has changed.

“We’ll treat Verizon fairly,” said Ms. Wiley, who leads broadband strategy for the mayor. “But where we have the power to make decisions, we will make decisions that benefit good corporate actors. They have to demonstrate to us that they are good corporate actors if they want us to use our discretion in ways that benefit them.”

The new approach also reflects how public the city’s battles with Verizon have become. In a recent Wall Street Journal story, Anne Roest, commissioner of the Department of Information Technology and Telecommunications, cited reports she had received of Verizon competitors having trouble running fiber-optic cable through conduits the company controls in Manhattan, Brooklyn and Queens.