Under a wrinkle that dates back to enactment of the law, members of Congress and thousands of their aides are required to get their coverage through new state-based markets known as insurance exchanges.

But the law does not provide any obvious way for the federal government to continue paying its share of the premiums for the comprehensive coverage.

If the government cannot do so, it could mean an additional expense of $5,000 a year for individuals and $11,000 for families under some of the most popular plans.

Not surprisingly, that idea is unpopular on Capitol Hill.

“It’s a very serious concern,” said Representative Billy Long, a Missouri Republican who said staff members were “freaked out” at the prospect of paying the full cost of insurance out of their own pockets.