In the general campaign, Mr. Romney has moderated his immigration positions as he tries to appeal to Latino voters, hoping to chip away at Mr. Obama’s big lead among that group. In the presidential debate on Tuesday, Mr. Romney endorsed proposals giving legal status to young immigrants who have been in the country illegally since they were children.

“The kids of those that came here illegally, those kids I think should have a pathway to become a permanent resident of the United States,” Mr. Romney said.

Without providing much detail, Mr. Romney said he would work with Congress on “real, permanent immigration reform” to give legal status to young immigrants. He has said that illegal immigrants who serve in the military should get permanent residency.

But Mr. Romney has criticized the temporary reprieves, which Mr. Obama created in June by executive action, as a political “stopgap measure.”

For young people who have grown up without documents, the deferrals and permits allow them to work legally and, in some states, obtain driver’s licenses and attend college at in-state tuition rates. In a recent poll by the Pew Hispanic Center, 86 percent of registered Latino voters said they approved of the program.

To qualify for the program, immigrants must be under 31, have come to America before they were 16 and have lived here for at least five years. They must also be current students or high school graduates. Since there is no filing deadline and no appeal if an application is denied, administration officials have urged young people to take their time to get it right. Many immigrants have also struggled to gather the papers they need and to raise the $465 filing fee.

Leading Republicans who are concerned about the party’s standing with Latino voters have differed on Mr. Romney’s position on the deferrals. In a recent interview on Spanish-language radio, Jeb Bush, the former governor of Florida, lauded Mr. Romney’s plan for broader legislation. But he said, “I think it makes all the sense in the world to maintain what exists right now.”