At the time, Herbert said the bill could lead to a "slippery slope" toward legal, recreational use.
Madsen's bill failed in the Senate by a single vote, although the lawmaker has said he will reintroduce it next year.
Connor Boyack, president of the Libertas Institute, a libertarian think tank, said he views the governor's comments as progress in the marijuana debate.
"Even during the legislative session, it became clear that the governor was backtracking his initial opposition to medical cannabis," he said, "so we expected this to happen and are encouraged to see him becoming more open-minded to it, just as many legislators are."
Boyack said there are widespread misperceptions about legal medical marijuana, and the group has scheduled a series of five town-hall meetings, the first scheduled for Tuesday, when the public and legislators are invited to discuss the policy.
"We completely agree with the governor that this policy should be well thought-out," Boyack said, "and we're spending an entire year doing just that so that when the Legislature takes up the issue again, a lot of the unresolved questions and concerns are resolved."
Herbert noted that Utah has legalized oils extracted from cannabis plants for treatment of people who suffer from seizures and other conditions and they appear to be beneficial.
"We have a history," the governor said, "of looking at opportunities for medical purposes to bring substances on board that maybe historically have not been traditional medicine."
He said whatever proposal Madsen brings forward next session will need to be the product of a collaborative process, working with state and federal agencies to ensure it can be executed and comply with existing laws.
"As a controlled substance, I don't have a problem," Herbert said.
"But the question is: Who's going to control it? Who's going to distribute it? What's going to be the quality and quantity and who makes those decisions? Those are questions that need to be answered if we're going to move ahead with medical marijuana."