The authorization would give some officers the power to enforce immigration law.
Backers, including Mayor Mark Bougton, say the training is needed because of the federal government's lack of action on immigration reform. The city has a large number of undocumented workers.
Opponents worry the training will result in racial profiling and will lead to a fear of police among immigrants, both legal and illegal.
Protesters outside carried signs and chanted "Stop 287," a reference to Section 287 of the federal immigration law that allows local police to form partnership with federal immigration officers.
Some city businesses, including Hispanic-owned restaurants, were closed Wednesday in protest.
The large crowd had dispersed by the time the nearly three-hour council meeting adjourned.
