Another idea is for the federal government to provide money to states to help pay the largest medical claims. Such assistance, which provides insurance for insurance carriers, has proved effective in reducing premiums in Alaska and Minnesota, and several other states will try it next year.

In addition, many Democrats say they want to provide more money to help consumers enroll in health insurance under the Affordable Care Act. Over the last two years, Mr. Trump has cut the funds for insurance counselors and enrollment assistance by 84 percent, to $10 million.

Reining In Drug Prices

Mr. Trump said he believed he could work with Democrats in Congress on “lowering the cost of prescription drugs,” and the Senate majority leader, Mitch McConnell of Kentucky, said the issue was sure to be on the agenda.

Democrats have praised two of the proposals Mr. Trump has advanced in recent weeks. One would require drug manufacturers to include the list prices of drugs in television advertising. The other would reduce Medicare payments for certain high-cost drugs by using the average of prices in other advanced industrial countries as a benchmark in deciding what Medicare should pay.

Drug companies oppose both ideas. They say the price disclosures would confuse consumers, who often pay less than the full list price. And drug lobbyists say Mr. Trump’s proposal for an “international price index” would just import price controls from other countries.

Lawmakers from both parties could also find common ground with the administration on a bill that requires manufacturers of brand-name drugs to make samples available to generic drug companies trying to develop inexpensive copies of those medicines.

Dr. Scott Gottlieb, the commissioner of the Food and Drug Administration, says some drug makers have tried to stifle competition by blocking access to samples.