The law requires that a Form I-9 must be filled out by every employee upon hire (or re-hire) and kept by the employer. The I-9 is the form that officially verifies that the employer has reviewed documents that prove that the hired person is eligible to work in the United States.
From time-to-time, a new I-9 form is issued by the government. For that reason, it is not recommended for employers to keep a folder full of copies to use, as it is easy to grab an outdated form. It is best to check online to be sure you are using the most current form.
The law states that employers must check work eligibilty AND fill out an I-9 which is retained. Many small business owners think that it is sufficient to just ask (or even worse, assume) that a person is work eligible. The law requires that this form be filled out for all employees. In the case of an audit, there can be a penalty for not having the forms filled out properly and available.
In addition, it is always more "professional" to have a "new hire procedure" for new employees. It inspires confidence in the company and allows the new hire time to ask questions. It is a good time to exchange information about goals, history of the company, workplace policies and procedures, and other practical concerns.
The forms and information can be found on the USCIS website. The page is called "I-9 Central" and is an excellent, user-friendly resource that has links to download the form as well as information about employee rights and a customer support link.
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