The proposal, known as a “safe third country agreement,” would potentially require asylum seekers transiting through Mexico to apply for protection in that nation rather than in the United States. It would allow U.S. border guards to turn back such asylum seekers at border crossings and quickly return to Mexico anyone who has already entered illegally seeking refuge, regardless of their nationality.

U.S. officials believe this type of deal would discourage many Central American families from trying to reach the United States. Their soaring numbers have strained U.S. immigration courts and overwhelmed the U.S. government’s ability to detain them. The Trump administration says the majority are looking for jobs — rather than fleeing persecution — and are taking advantage of American generosity to gain entry and avoid deportation.

“We believe the flows would drop dramatically and fairly immediately” if the agreement took effect, said a senior Department of Homeland Security official, speaking on the condition of anonymity to discuss negotiations with the Mexican government, which the official said had gathered momentum in recent weeks.