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An employer does not need consent to conduct background checks unless it hires a third party background check service to perform the background checks. In that case, the Fair Credit Reporting Act does require employers to get consent, to disclose a copy of the report to the employee, and to specify what information contained in a report is a basis for adverse employment action if adverse employment action is taken based on the contents of the report. See here for more information.
If the employer is conducting the background check on their own, the requirements of the FCRA do not apply. Employers are simply restricted from inquiring about arrests that did not result in conviction, , convictions more than 7 years old, or any criminal record that has been expunged, sealed or dismissed. Credit checks are the one exception. Those always, require employee consent, and in fact the law in CA actually prohibits credit checks except for certain kinds of employees. See here for more information.
As for telling employees what will be grounds for termination, there is no need to state this up front, and in fact doing so can restrict your ability to use discretion. It is best to decide things on a case by case basis.
Lastly, if an employee refuses to consent to a background check, that is grounds for termination in an at will employment state. In other words, while the employee has a right to refuse a commercial background check, an employer is not forced to keep that person employed. Refusing a background check is a huge red flag and a perfectly reasonable reason to let someone go.
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