There is no absolute ban on hiring persons with felony records into such positions. Employers generally will consider the issue among other factors in the employer's application. A complete bar
on persons with criminal histories can result in a discriminatory impact in that certain racial groups end up being excluded more often than others. In an analysis of a single applicant there is nothing preventing an employer from refusing employment on the basis of a criminal history.
In this instance, the conviction record may have been sealed or expunged in which case this employer will not consider that criminal history. Your employer has determined there is no threat or failed to receive all needed information to make an informed decision.
If the problem is the employer has determined there is no threat, then there is little that you can do. If the employer has made a decision based on insufficient information, then you may have recourse. You may obtain the information yourself and submit those records to the employer anonymously. You may also bypass human resources or compliance if you believe they are ineffective and submit your concerns in writing to management. Ultimately, this issue is entirely left to the employer's discretion. This will change if fraud or mishandling of funds actually does occur.
It appears that your company handles Department of Education funds. As such, you may report suspected mishandling of those funds to the Office of Inspector General at:
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