It would prevent landlords from reporting tenants to immigration authorities.
Some advocates for immigrants say landlords use the threat to get undocumented tenants to pay higher rents or vacate their units.
Protesters in downtown Los Angeles said there is a trend of harassment against tenants based on their immigration status.
"There's a huge housing crisis and a lot of landlords want to capture the value of the units," said Estuardo Mazariegos with the group Californians for Community Empowerment. "So they're threatening long-time residents in rent-controlled units with immigration status if they're undocumented."
The advocates want the governor to sign AB291. by Assemblyman David Chiu, D-San Francisco. It would prohibit landlords from disclosing the immigration status of tenants or threatening to report them to immigration authorities.
The state Assembly on Monday voted to send the bill to Brown.
Brown has not yet indicated if he intends to sign the bill.