The measure, Assembly Bill 1840, was first introduced in mid-January, and after several amendments, it advanced last week to the Committee on Housing and Community Development, where it awaits further action.

Assembly Bill 1840 would change existing law to allow illegal immigrants to be eligible for the California Dream for All Fund, which provides interest-free loans for a down payment on a home for first-time buyers.

The bill was introduced by California Assemblyman Joaquin Arambula, a Democrat, who last month told GV Wire, a Fresno-based news outlet, that he "wanted to ensure that qualified first-time homebuyers include undocumented applicants."

Last week, as the bill advanced to committee after amendments, Mr. Arambula told the Los Angeles Times that, historically, homeownership has been the main way people accumulate generational wealth in the United States.