WASHINGTON — As more states begin setting up health care exchanges for individuals to buy insurance under the new health care law, abortion opponents are using the opportunity to try to restrict abortion coverage in private insurance plans.

Most Americans with employer-based health insurance currently have coverage for abortion, according to the Guttmacher Institute, an organization that researches reproductive health and rights — a fact that few people seemed to know until the health care debate last year.

It even eluded employers. In a 2010 survey of employers, the Kaiser Family Foundation found that 71 percent of employers who were interviewed did not know whether the main health insurance plan they offered covered elective abortions.

Now the coverage is getting increased scrutiny as states start to build their own systems under the new law, and a growing number of states are passing bans.