The debate comes at a complex moment for opera, as the art form begins its fifth century.

Some American companies have suffered, including New York City Opera, which closed last year, and the San Diego Opera, which survived a brush with death this year. Lyric Opera of Chicago recently announced that its ticket sales rose last season, but that its biggest seller was not an opera but a musical, “The Sound of Music.” Some European companies, including the Vienna State Opera, which gets a great deal of government money, sell nearly all their seats. But others have difficulties: English National Opera, which has been riding a wave of critical success, just had its public grant cut by Arts Council England, which wrote that the company had “struggled to reach box office targets and to achieve long-term stability.”

The Met, which can hold 4,000 operagoers and stages more than 200 performances each season, faces its own challenges. Mr. Gelb said that the company must reduce its labor costs to survive, but that he was willing to consider alternatives to his proposals to the unions.

“No cuts means no Met,” he said in a recent interview.

The Met’s box office problems in recent years have led it to sell more seats at a discount to fill the house and to lure new audiences. During at least seven performances last season, more than 1,000 seats — roughly a quarter of the house — were sold at a discount, according to box office data the Met shared with unions that was obtained by The New York Times. The average discounted tickets at those performances sold for between $37 and $53, well below that average full-price ticket at the Met, which was $156 last season.

The Met said that it had earned 73.2 percent of its potential box office revenue last season — up from the 69.1 percent it earned the season before, but still well below what it took in during the 1990s, when the figure hovered around 90 percent. The Met said that the majority of its discounted tickets were sold through marketing offers like those used on Broadway, and were aimed at specific groups — including first-time operagoers, readers of certain websites or publications, tourist organizations and cultural groups.

It is against that backdrop that the company is seeking significant concessions from its unions for the first time in decades. The contracts of 15 of the Met’s 16 unions expire on July 31, and some unions warn that next season could be delayed by the dispute. The Met’s proposals would not cut basic salaries, but would alter the work rules that govern life backstage, effectively cutting take-home pay. The dispute has centered on not only whether to change work rules, but on what impact the proposed changes would have.