FAIRFAX, Va. — Jerry Pinto, an immigrant from Bolivia, has visions of opening a spacious carpentry workshop in this suburban city, with his name in bold letters over the door.

“I want a place where I can be visible,” he says wistfully. But for now he knows he has to lie low, because he is in the country illegally. He runs his carpentry business almost surreptitiously from the cramped garage behind his house.

Mr. Pinto is among more than four million unauthorized immigrants whose lives could be transformed by the Supreme Court. On Monday, the justices will hear oral arguments in a challenge brought by 26 states, led by Texas, to President Obama’s effort through executive action to give the immigrants legal work permits and protection from deportation.

Depending on the outcome, people like Mr. Pinto will have a chance to come out into the open or will remain, perhaps for years, in a twilight underground. And the stakes are high in this election year, since the two leading candidates for the Republican presidential nomination, Donald J. Trump and Senator Ted Cruz of Texas, have both said they would deport all 11 million foreigners in the country illegally.