Senator John J. Flanagan, a Long Island lawmaker who leads the Republican minority, recently said that passing the bill would be a “colossal political mistake” for Democrats. Nick Langworthy, the incoming chairman of the state Republican Party, called the bill a product of the “extreme left” that showed disdain for the rule of law.

County clerks have again denounced the proposal, with some vowing to defy it if it becomes law, and county sheriffs have warned that it would constrain their ability to enforce traffic safety.

The issue is so fraught that even some who publicly support the bill have privately worked to block it. At a fund-raiser earlier this month on Long Island, Mr. Cuomo and Jay Jacobs, the leader of the state Democratic Party, met with five suburban senators and told them that voting for it would be politically perilous, according to two people familiar with the conversation.

Mr. Jacobs, a close ally of Mr. Cuomo, would not comment on that exchange. But he said that several senators as well as Assembly members had asked for his opinion on the issue, and that he had advised them not to heed the demands of the “far left.”

“I think there is a measure of arrogance in forcing people to accept things that they’re not willing to,” Mr. Jacobs said. “Those people, particularly in the city, frankly, who are pushing this, are really shortsighted in my view.”

He added that although he said he supported expanding licenses in theory, he would rather “play the long game” and not risk the Democratic majorities in both chambers .

None of the six Democratic senators from Long Island returned requests for comment.

Robert Mujica, the governor’s budget director, said he was present at the Long Island meeting and denied that the governor had discouraged senators from voting for the proposal.