If Mr. Trump picks Ms. Verma to succeed Mr. Price at the Department of Health and Human Services, it would be taken as a sign among many that he wants to continue vigorous opposition to the Affordable Care Act, with the government doing the minimum required by the law to implement its provisions. Ms. Verma, an ally of Vice President Mike Pence, worked closely this year with Republicans in Congress on their proposals to undo the law and to cut Medicaid, the program for more than 70 million low-income people.

Still, some progressives have interpreted her work under the health care law in Indiana, where Mr. Pence was governor, to mean that while she opposed the Affordable Care Act, she was committed to finding ways to enforce it if it remained on the books.

Mr. Gottlieb has more experience in Washington and was seen at the time of his appointment as the more moderate of candidates being considered. In his first months at the F.D.A., he has deftly balanced the concerns of patients and pharmaceutical companies, while taking steps to combat the opioid epidemic and speed access to lower-cost generic drugs. His nomination would be seen as a signal that the president might want to take a different approach to the health care debate.

“We have the votes on the substance but not necessarily on the process, which is why we’re still confident that we can move health care forward and get it done in the spring,” Sarah Huckabee Sanders, the White House press secretary, said before Mr. Price’s resignation.

After the latest legislative failure, Mr. Trump said he would sign an executive order in the coming week intended to enable Americans to buy health insurance across state lines, a sign that he did not intend to wait for Congress. But it is not clear that he has the authority to do that on his own, and states often resist federal efforts to intrude on their regulation of insurers.

Senator Lamar Alexander of Tennessee, the Republican chairman of the Senate health committee, and Senator Patty Murray of Washington, the panel’s ranking Democrat, have resumed negotiations on bipartisan legislation intended to shore up the current insurance exchanges and prevent prices from shooting up.

The uncertainty comes at a crucial moment, just as federal and state officials are preparing for the fifth annual open enrollment period under the Affordable Care Act. The open season, when people can sign up for coverage, runs from Nov. 1 to Dec. 15. Critics say that the Trump administration has destabilized insurance markets, driving up premiums for 2018 and making it harder for people to enroll.