Last Night in Twisted River by John Irving, there is a surprise deal struck by two of the main characters. I won't spoil it for those who have not yet read the book, but for those who have, they know the incredible agreement I am writing about.
It occurred to me, as I read, that negotiations are possible in almost all aspects of life -- from reducing mortgages to reducing criminal sentences. Not surprisingly, then, it is also possible to negotiate with the Department of Homeland Security when a foreigner is deportable.
In fact, several programs exist in the regulations allowing DHS to negotiate with a foreigner in just this way. The most well-known is Deferred Enforced Departure. Everyone admits that the foreigner must leave, but DHS is willing to officially place a low priority on the removal, usually for humanitarian reasons. There is also Extended Voluntary Departure, which grants safe haven in a way similar to Temporary Protected Status. TPS is a designation by the Attorney General permitting persons from a specific nationality to remain.
Because it is never known whether a person will be able to benefit from these programs in the near future (as is the case of Chileans currently in the U.S. without authorization -- no TPS designation yet despite the earthquake), I rarely advise clients to leave the country voluntarily.