American citizen as a sponsor for overstayed immigrant?

In general, an alien has to be in a lawful immigration status to change to another lawful immigration status. In general, an alien without legal immigration status cannot be approved for an employment-based immigration status.

The government is starting to crack down on employers who hire aliens without work authorization.

You should review your specific facts with your attorney to see what options you have. You should tell your worker to review the worker's facts with the worker's attorney.

As my colleague suggests, there is often not an easy path for someone who has overstayed their nonimmigrant visa. The question is whether what you propose is impossible or merely challenging. The answer will depend on the type of visa status that the employee was in previously, whether he or she had begun an immigration-related process in the past, and what "equities" the person may have in the U.S. You as an employer should heed my colleague's warnings and determine what exposure you may have to continuing the employment of someone known to you as an "illegal worker", which is a term that I do not personally favor. Both you and the employee would do well to obtain the best possible advice. There are many great immigration attorneys in Miami, but you would not go wrong to give a call to Ira Kurzban's office.

As a general principal, an employer may sponsor an employee for a green card. It is unclear form your question, whether the worker overstayed their visa or entered illegally. If they overstayed their visa, it will depend how long they accrued unlawful presence. If they entered illegally, I would not risk trying to sponsor them.