LOS ANGELES — Labor unions across the country have been among the most vocal and important backers of the nationwide fight to increase the minimum wage to $15. At the same time, some labor leaders — including one here who helped lead the successful effort to raise the minimum wage in Los Angeles — have encouraged cities to adopt provisions that would allow employers with unions to pay below that level.

Last week, when the City Council here voted to raise the minimum wage to $15 an hour from $9 an hour by 2020, the Los Angeles County Federation of Labor, an umbrella group of local unions, brought hundreds of workers to voice their support for the proposal.

But this week, the same group was back before the Council asking that union workers be exempted from the higher wage requirement, saying that such a provision would give business owners flexibility to offer more generous health insurance packages, for example, and to make sure there was no conflict between federal labor laws and local minimum wage ordinances.

The plea from the union group drew criticism from many elected officials who had voted for the wage measure, as well as some accusations of hypocrisy. Critics argued that the Los Angeles County Federation of Labor was simply trying to increase its membership by giving employers a financial incentive to agree to a union contract as a way to gain exemption from the higher minimum wage.