It depends on the jurisdiction in the United States. Some States and or local jurisdictions limit who can become police officers, limiting the postion to only US Citizens. You should contact the human resources department of the police department you wish to join and ask if they allow LPRs to become police officers. Good luck.
Agreed, it depends on the jurisdiction where you would like to become an officer. The easiest way to find out would be to call the PD in your area and ask them what the hiring requirements are.
I agree with my colleagues. You need to contact the police department where you would like to apply and ask them if permanent residents can serve as police officers.