This is similar to, but somewhat different from, the usual welfare objection to immigration. The claim is that by increasing labor supply, immigrants push down the wages of natives and may push some natives to unemployment. The lowr wages and increased unemployment among natives leads to the natives making greater use of the welfare state.
Also related: second-order harms to immigrant-receiving countries and second-order crime.
A version of this argument was made in the paper Immigration and African-American Employment Opportunities: The Response of Wages, Employment, and Incarceration to Labor Supply Shocks (NBER page) by George J. Borjas, Jeffrey Grogger, and Gordon H. Hanson.