“This is a very big deal,” said Paul K. Sonn, the general counsel to the National Employment Law Project, a national research and advocacy group on wages. “It would mean a raise for one of every three workers in the state.”

The movement has gained traction in a dozen cities, including Los Angeles, San Francisco and Seattle. Last week, the mayor of Washington proposed raising the minimum wage to $15 by 2020.

In New York, Gov. Andrew M. Cuomo is pushing for a $15 minimum wage in New York City by the beginning of 2019 and statewide by July 2021. Mr. Cuomo, a Democrat who has previously raised wages to $15 for state workers and fast-food employees, was attempting to broker a deal with the State Senate, which is led by Republicans who remain wary of the potential economic effect of the wage, particularly in long-suffering areas upstate.

The deal comes as raising the minimum wage has become a central point of contention in the Democratic presidential contest. Bernie Sanders, the Vermont senator running for the Democratic nomination, has made a $15 federal minimum wage a central plank in his campaign; it is now $7.25 an hour.

His opponent, Hillary Clinton, has called for a $12 federal minimum wage, but has said she supports efforts in some cities and states to pass a $15 minimum wage. She has resisted pressure to support a $15 federal minimum wage.