When Ashley de Azevedo married in 2012, she knew that her U.S. citizenship would make her husband, an immigrant from Brazil, eligible for a green card. What she didn’t realize was that to obtain permanent residency, he would need to return to Brazil for 10 years because he had entered the United States illegally.

“It was a devastating reality,” Ms. Azevedo, 38 said. “I was pregnant, and he would miss out on years of our child’s life.”

So Mr. Azevedo stayed in the United States, vulnerable to deportation but with his wife and his son, who’s 12.

Now, a policy under consideration by the Biden administration could provide Mr. Azevedo and other undocumented spouses with a path to permanent residency that would not force them to leave the United States.

Calls for such a move have been growing in some quarters and could give President Biden a political boost in battleground states with large immigrant populations. But the idea is drawing sharp criticism from some Republicans and immigration hawks, who regard it as an abuse of executive authority.

Word of the proposal came just days after the administration took long-expected action to make it much harder to seek asylum, which had become an all but certain path for remaining in the United States and helped drive record levels of migration in recent years.