“It is not a priority,” said Joseph Geiger, the executive secretary-treasurer of the New York City District Council of Carpenters, which spent $750,000 on campaigns last year, records showed.

In interviews with a dozen unions, only two — Local 32BJ, which represents janitors, doormen and airport workers, and the local arm of the Communications Workers of America — offered unqualified support for public financing. Two others were vocally opposed. The rest declined to comment, took no position or declared other matters to be of greater importance.

A public financing system would typically lower the maximum contribution to candidates, but then offer a generous matching-fund allotment. In New York City, for example, candidates can receive an eight-to-one match of public funds to small donations.

Those who favor such a system say it lessens the advantages of incumbency by reducing the impact of large donors. But for some of those large donors, including unions, that might be a reason to oppose it.

When the Supreme Court was considering the landmark Citizens United case, which found that government cannot ban corporate political spending, the national A.F.L.-C.I.O. filed a brief supporting corporations. The union argued that looser restrictions on political spending would help labor unions, although it later said that the ruling went too far and has urged the case be overturned.

In New York this year, when a coalition of activists fighting for campaign finance reform, which included the two supportive major unions, reached out to other unions to ask for their support, they were rebuffed, said a person familiar with the coalition’s efforts, who spoke on the condition of anonymity to protect their projects with labor unions.

“If we’re not there, then it would be slanted in a direction that is much different than the unions,” Mr. Mulgrew said, adding that his union’s donations counteract the influence of right-leaning groups.