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Verizon workers rallied in New York Saturday, calling on the company to negotiate a more generous contract. Reuters/Eduardo Munoz

Many of the almost 40,000 workers at Verizon Communications Inc.'s wireline unit voted to strike amid negotiations over a new labor agreement. The Communications Workers of America (CWA) union announced the results of the vote -- 86 percent of the workers who cast ballots approved the strike -- during a rally in New York Saturday.

The move does not necessarily mean workers will go on strike. It is designed to give negotiatiors with the CWA and the International Brotherhood of Electrical Workers more power at the bargaining table and increase pressure on the telecommunications giant to reach an agreement before the current four-year labor contract expires Aug. 1. As those talks proceed, unions have slammed what they call Verizon's demands to "eliminate job security, slash pensions and increase health care costs."

Verizon representative Rich Young blasted the protest. "Union rallies and strike-authorization votes are useless distractions that achieve nothing," Young told Reuters. "We believe their time would be far more beneficial focusing on the important contractual issues that need to be resolved."

In addition to union leaders, New York City Public Advocate Letita James spoke at the rally:

The contract being negotiated covers 39,000 workers in nine states along the East Coast and Washington. It includes employees who work in landline telephone operations and the popular FiOS network, which bundles together Internet, phone and television services.

Union membership at Verizon has declined considerably over the past couple of decades -- the combined result of buyouts, retirements, technological changes and the company's sales of certain operations. Less than one-third of the firm's employees are now covered by collective-bargaining agreements, and Verizon's wireless unit remains effectively union-free. At the same time, labor strife at the company has proven to be a relatively common occurrence. Verizon's landline workers last walked off their jobs en masse in 2011 as negotiatiors battled over the terms of the current agreement.

If the workers do go on strike, it would be the largest work stoppage in the U.S. this year.