It depends on the state if you are talking about the US, for example for the LAPD requires you to be a citizen or to have applied for citizenship. Being a permanent resident isn't enough.
The City of Los Angeles requires that a Police Officer candidate be a United States citizen, or that a non-citizen be a permanent resident alien who, in accordance with the requirements of the U.S. Citizenship and Immigration Services (USCIS), is eligible and has applied for citizenship.
During the selection process, each non-citizen is required to prove that USCIS accepted his/her application for citizenship prior to the date the Police Officer written test was taken.
California State law requires that citizenship be granted within three years after the employment application date. For information regarding citizenship requirements, contact the USCIS.