PITTSBURGH—The push by U.S. manufacturers to cut labor costs, especially for health care, is deepening the divide between three of the biggest U.S. steel companies and their main union. Contract talks, which are usually resolved by November, have dragged on months after their summer deadlines.

Premium-free health-care packages are one of the last big perks awarded by the top tier of American manufacturers, including auto and steelmakers. But now, slowing demand for base materials, along with the rising cost of health care,...