Glendy Martínez is waiting anxiously to see if Congress will ever pass legislation to allow immigrants like her, without papers, to stay in the country legally. But frankly, she says, she does not care if it will include any promise of citizenship.

With the earnings from her job in a Houston hair salon, Ms. Martínez, 30, is supporting one child born in Texas and three others she left behind in her home country, Nicaragua.

“So many people back there depend on those of us who are here,” she said. “It would be such a help if we could work in peace and go back sometimes to see our children.”

As President Obama looks for a way to salvage a broad overhaul of the immigration system, he opened the door this week to a piecemeal series of smaller bills as a way of getting past the objections of the Republican-run House, which refused to take up the comprehensive measure that the Senate passed in June.