New York City expects its two most popular employee health plans to reach taxable Cadillac levels by 2018 or shortly after. This year, the city projects that it will pay a total of $7,128 for individuals and $18,249 for families in its most popular plan, including the costs the city pays into union welfare funds to cover prescription drug benefits. That is above the national average for employer-sponsored health care coverage, which last year was $5,615 for single coverage and $15,745 for family coverage, according to a 2012 Kaiser Family Foundation survey.

The total health care cost for the city’s nearly 300,000 municipal employees, pre-Medicare-age retirees and their dependents is expected to approach $8 billion by 2018.

In a letter in April to the head of a labor coalition, Caswell F. Holloway IV, deputy mayor for operations, said the Cadillac tax would cost New York City $22 million in 2018, increasing to $549 million in 2022. (This year, the total city budget, excluding federal and state aid, is just over $50 billion.)

“We know that, on the current trajectory, we’re going to be hit with that tax and it would increase very steeply,” Mr. Holloway said.

So the administration of Mayor Michael R. Bloomberg, in its final months in office, is asking municipal unions to agree to seek new bids for the city’s health insurance business, hoping to lower premiums. It has already achieved one small victory, getting the city’s current primary insurer to freeze premiums for one year if it keeps the city’s business, the mayor said on Friday.

But lower-cost plans are likely to involve greater out-of-pocket costs and more limited networks of doctors, and so far, the response from labor has been cool.

Ninety-five percent of city employees and 93 percent of retirees are in the two largest plans, which require employees to pay nothing toward their premiums. According to the Kaiser Family Foundation survey, the average contribution by public employees throughout the country is 12 percent for individual plans and 23 percent for family plans.