They point to a 2018 study that showed an association between marijuana use, at least daily, and remaining in medication-assisted treatment. Another study, from 2014, found that states with medical marijuana programs had lower death rates from opioid overdoses than states without.

“What we’ve come to understand is that marijuana in many instances is an exit drug, not a gateway drug,” State Representative Renny Cushing, a Democrat and the lead sponsor of a bill being considered in the New Hampshire House, said.

But even those experts who believe that marijuana may have a role in helping people with opioid addiction say that much more research is needed, and that marijuana should not be a substitute for treatment with methadone or buprenorphine, which have been found to substantially reduce deaths among people addicted to opioids. Recently, some states have begun encouraging people to switch from opioids to medical marijuana, or even making opioid addiction a qualifying condition to get medical marijuana — a move that some experts have found alarming.

Mr. Cushing’s bill would make it legal for people 21 and over to possess, consume, buy and grow small amounts of marijuana, and would create a commission to license and regulate marijuana cultivation, production and retail establishments. Taxing marijuana sales could bring New Hampshire between $15.3 million and $57.8 million a year, a state commission has said. The bill, which is likely to be voted on in a House committee on Thursday, is expected to pass the House; if it passes the Democratic-held Senate and reaches the governor’s desk, Mr. Sununu, who declined to be interviewed for this story, has promised to veto it.

Some people suggest that opponents are using New Hampshire’s opioid crisis to back up their argument, when their real reason for opposing legalization is a more general discomfort with drugs of any kind.

Keith Howard, the executive director of Hope for New Hampshire Recovery and a member of the governor’s commission on alcohol and drug abuse, said that he was neutral on the bill.

The opioid crisis and the question of whether to legalize marijuana “have nothing to do with each other,” he said, though “in debate it’s very easy to conflate the two and then use the power of deaths from opioids to bolster an argument about marijuana.”