But a hiring spree of immigration judges under President Trump could make the process faster and potentially more effective, so that alternative is still on the table.

The final three ideas under review — making it harder to apply for asylum, extending the use of GPS ankle monitors, and speeding up enforcement of deportation orders — also carry potential pitfalls.

Ankle monitors are typically used to track migrants for a few months after they cross the border. The new proposal would require the devices to be worn for the full duration of immigration cases, which take about two years to decide on average — a very long time to have to wear such a device. The proposal would require more funding, since it would require a much larger number of ankle bracelets.

The final proposal calls for changing current deportation procedures, under which migrants are given a date by which they must leave the country or report to an immigration office for deportation. Homeland Security officials cautioned that trying to deport people immediately could leave some children stranded — if their parents leave home for court, for example, and never come home. The policy could also further deter people from showing up to court.

In an interview at the height of the family separation controversy over the summer, Stephen Miller, the president’s senior adviser on immigration issues, said that voters would support efforts to crack down at the border. “I have absolute confidence, as does the entire administration, that the American public wants us to have a fully secure border with predictable consequences for illegal entry,” he said.

One way or another, the arrival of thousands of new migrant families will inevitably affect political races in border states like Arizona. These days, the Greyhound bus stations in Tucson are overflowing with new arrivals, and shelters are so full that some migrants are being checked into local motels.

“Every day, every bed has been taken in the last couple weeks,” said Gretchen Lopez, who runs the Inn, a migrant shelter in the basement of a Methodist church in Tucson. “I see no indication that it is slowing.”