New York is close to joining a growing list of states that are creating a retirement plan option for private sector employees who do not have access to 401(k)-like programs at work.

The New York program was included in Friday’s budget deal, which Gov. Andrew M. Cuomo is expected to sign. The plan would enable businesses to provide workers with access to Roth individual retirement accounts overseen by the state. An estimated 3.5 million private sector employees in New York work for employers that do not offer a pension, a 401(k) plan or another savings option, according to AARP, which has lobbied in support of the plans.

“For years, we have been working to develop and pass a retirement program that would give millions of New Yorkers the opportunity to save for their futures,” Mr. Cuomo said in a statement. “In this year’s budget, we proposed and passed a common-sense, progressive reform that will strengthen our work force.”

Ten states, including New York, are on the verge of enacting such plans or already have them. The plans have been moving ahead even though Congress rolled back Obama-era rules meant to encourage states to create retirement programs for people without workplace savings accounts.