Republicans used last year’s tax bill to remove the law’s penalty on people who do not secure insurance. With two more years, party officials say, they could try to eliminate the so-called employer mandate, removing penalties for employers who do not offer coverage for their employees.

Others would push lawmakers to try to go further and cut at the heart of the law’s popularity: its protections of those with pre-existing conditions.

“Republicans need to grow a spine and confront that debate head-on,” said Mr. Roth, who believes Democrats have scared Republicans into abandoning their position that the “heavy federal mandate” is driving up insurance costs.

Find common ground or head rightward?

With a presidential election looming, Republicans also face a consequential choice: work with Democrats to achieve some policy victories and lift Mr. Trump, or take advantage of what could be a dwindling period of unified Republican control to press rightward.

Lawmakers in both parties believe there is at least a chance that a Republican House and Senate could pass bipartisan legislation overhauling incarceration laws and prison sentencing, which would include changes to mandatory minimum sentences and programs meant to reduce recidivism.

“The first thing you need to be is gracious,” said Senator Lindsey Graham, Republican of South Carolina. “Thank you very much to the American people for allowing us to stay in charge. You need to challenge the Democrats to find common ground.”