A union representing 1,200 U.S. air maintenance workers at United Parcel Service Inc. turned up pressure on the company Sunday to settle a three-year contract dispute, saying it would seek clearance to strike.

The union is taking its grievances directly to UPS shareholders, running as an advertisement an open letter to David Abney, the company’s chief executive, ahead of a Thursday shareholders meeting.

The letter, which has been delivered to board members, was signed by nearly 78 percent of members of Local 2727 of the Teamsters union, asking the company to maintain air mechanics’ current health plan and not demand other concessions.

“We’re not willing to back off of this and we will strike over it,” Tim Boyle, the local president, said.

The company said that it continues to negotiate in good faith with the union.

“Talks continue under the control of the National Mediation Board, which has scheduled sessions several months out,” Mike Mangeot, a spokesman for UPS Airlines, said in a statement.

“The union’s talk about a job action is simply posturing and a common union tactic designed to pressure talks. Our mechanics are good people who do a good job of keeping our aircraft flying safely and reliably, and UPS continues to negotiate in good faith for an agreement that’s good for them, the company and our stakeholders,” he said.

Union members also will protest at the UPS shareholders’ meeting Thursday in Wilmington, Delaware, with protests outside the meeting and, for union members who are also shareholders, questions to company officials inside.

The local plans additional protests Tuesday in Atlanta, where the company is headquartered.

The union already voted in November to strike, but it saw that request denied by federal authorities. The air maintenance workers are governed by the U.S. Railway Labor Act, which only allows strikes after it finds negotiations and mediation have failed.

But if the company does not agree to keep members’ health plans intact at the next bargaining session, on May 11 and May 12, Boyle said the union would ask again for permission to strike.

Even if the board grants permission, though, a strike would take at least another 30 days because of other procedural hurdles.

A strike could ground the package delivery company’s airplanes and disrupt packages sent by air, even as UPS and its rivals grapple with higher costs for surging e-commerce business.