It was Mr. Hutchinson’s Democratic predecessor, Mike Beebe, who got the state’s Republican-controlled legislature to expand Medicaid by using federal funds to buy private insurance for the poor. Mr. Hutchinson said in the interview that he wanted to continue the expansion here, but that if the federal government stopped paying most of the cost as the health law requires, “we’ll just have to look at it again.”

He said he would either seek the Trump administration’s permission to impose a work requirement or seek a block grant — and, with it, leeway to create new rules — to cover those who newly qualified for Medicaid under the health law.

As a rough model, Mr. Hutchinson pointed to a work requirement for certain food stamp recipients — a federal policy that was part of the landmark 1996 welfare overhaul — that Arkansas reinstated last year. Under that rule, able-bodied adults without dependents cannot receive food stamps for longer than three months unless they are working, volunteering or getting job training for 20 hours a week.

States were allowed to suspend the requirement during the recession, but most have brought it back as unemployment has dropped. It generally applies to adults ages 18 to 49 unless they are pregnant, have dependent children or are medically certified as “unfit for employment.”

Since Arkansas reimposed the work requirement last year, its food stamp rolls have lost about 36,000 people, according to the State Department of Human Services. Mr. Hutchinson said the drop was partly explained because people had found jobs, but advocates for the poor said many appeared to have been cut off.

For now, Arkansas is sending letters to Medicaid enrollees to let them know they are eligible for “free job search assistance,” including career counseling and help writing résumés. The state is putting together a system to keep track of those who take advantage of the offer and whether they end up finding jobs. Many Medicaid recipients here already have low-paying jobs, but roughly 40 percent report having no income, according to the state.