That requirement, known as the individual mandate, is one of the more unpopular features of the law, signed in 2010 by President Barack Obama. But insurance companies like it because it requires people to buy their product, bringing in healthy people who pay premiums and do not use much care.

Analyzing the Republican strategy, Joel L. Michaels, a health lawyer at the firm McDermott Will & Emery, said there was “a tension” between efforts to repeal the health law and shore up its insurance marketplaces, where more than 10 million people obtained coverage last year.

“A political agenda premised on the Affordable Care Act being unworkable could conflict with efforts to support the A.C.A. exchanges, even on an interim basis,” Mr. Michaels said. “How far do you go with short-term fixes, which could make the law work better in the long term? It’s a delicate political dance.”

Insurers are seeking immediate governmental action because they must decide by early May what kinds of health plans they will offer on the exchanges in 2018.

The proposed rule drafted by the Trump administration and one of the bills drafted by House Republicans would make it more difficult for consumers to obtain insurance outside the annual open enrollment period. Consumers would have to provide documents to show they were eligible for a special enrollment period. Under existing rules, people can sign up after the deadline if they experience certain “life changes” like having a baby, getting married, losing employer-sponsored insurance or moving to a new state.