CATHEDRAL CITY, Calif. — Faced with losing the life they’ve built together in the dusty California desert town of Cathedral City, Doug Gentry and Alex Benshimol are making a last-ditch effort to stave off the looming threat of deportation.

To a large degree, the couple is stuck. While the American information technology consultant and Venezuelan pet groomer wed at a romantic Connecticut ceremony last year, the federal government won’t recognize the marriage between the two men — and as a result, won’t approve their application for a green card.

But the couple, and others facing a similar predicament, are still trying. The men don’t expect to actually obtain a green card any time soon and have already been shot down once but hope filing an application might convince an immigration judge to at least refrain from deporting Benshimol while the fiery legal debate over the country’s same-sex marriage laws simmers.

For years, immigration attorneys warned gay couples not to bother seeking a green card for their foreign spouses since there was no chance they’d get one. Now, in select cases, they’re starting to rethink that advice.

In the wake of the federal government’s announcement that it will no longer defend a law that defines marriage as between a man and a woman and a court ruling raising questions about the law, some immigrant advocates have suggested that gay couples fighting deportation apply for a green card in a final effort to stay in the country.

But the small group of couples already facing deportation has little to lose by applying, and might see some gain.

In March, an immigration judge in New York halted deportation proceedings involving a lesbian couple until December. Last month, an immigration judge in New Jersey did the same for a Venezuelan salsa dancer married to an American graduate student after Attorney General Eric Holder asked an immigration appeals court to review another case involving a same-sex couple.

In a memo posted to its website in March, the American Immigration Lawyers Association suggested that couples facing deportation consider filing for a green card in the hopes that it might win sympathy from an immigration judge willing to put the case on hold or bolster the immigrant spouse’s case for an asylum petition.