There are about 500,000 so-called fugitive aliens who have received deportation orders. If they come forward in August, those illegal immigrants are able to have about 90 days to organize their affairs without upheaval.
"They may own property, they may have vehicles, clearly they may have children in school ... Again, this will give them the opportunity to work with ICE, and to close out their affairs, before they comply with the law and return to their native countries," said Virginia Kice, Immigration and Customs Enforcement.
Immigrant advocates say sigining any document, including one for voluntary departure, would mean giving up chances of becoming a legal citizen.
"This is not a solution. A real solution would actually allow us to stay in this country, with our families, with our jobs, and actually continue to contribute to the strengthening of our country," said Angelica Salas, Coalition for Human Immigrant Rights.
There are holes in the system. Some fugitive aliens may not be aware they are on the deportation list because their papers have been mishandled by unlicensed paralegals, or "notarios."
"Attorneys can file a motion to reopen based on lack of notice, based on fraud from a notario, even based on fraud by an attorney," said immigration attorney Gustavo Mora.
In addition, some say returning immigrants to their native countries would be a big step backward.
"Because if you go home, you have no food, no opportunities to work. That is no money," said Carmen Garcia.
Immigration courts are backlogged and detention centers are full. Federal officials hope the program will ease the pressure.