“You can’t stand in the way, because there’s other delivery methods that are effective,” said State Senator Diane J. Savino, a Democrat who represents parts of Staten Island and Brooklyn and who is the sponsor of that chamber’s bill. Ms. Savino added that the governor “gave a lot of that we wanted.”

The Assembly has passed bills in support of medical marijuana five times, most recently in May, only to see the measures die in the Senate, where Republicans were cool to the idea.

In recent weeks, however, several Republicans joined Democrats in voicing support for Ms. Savino’s bill.

The Senate Republican leader, Dean G. Skelos, said the bill would be voted on and passed on Friday.

On Thursday, parents and other supporters — including some with children in wheelchairs, and others visibly with degenerative conditions — pleaded for the bill’s passage. “Please pass this act,” said one parent, Tim Emerson, whose 7-year-old daughter, Julia, has epilepsy. “Please help our kids.”

Despite acknowledging its emotional pull, Mr. Cuomo said that he was wary of allowing marijuana to become too widely or too easily available. In recent days he said he feared that it was “a gateway drug,” and observed that the state was already dealing with a resurgence of heroin use.

On Thursday, however, Mr. Cuomo seemed to hold out the possibility that adjustments could be made by the Health Department in the future to allow for changing circumstances, including an increased need for the drug.

“It’s a knob: You can turn it up, you can turn it down,” he said, by tinkering with the number of dispensaries or diseases for which treatment with marijuana is permitted.