MEXICO CITY — When American and Mexican officials prepared to meet in Guatemala in July, one issue in particular was weighing on the United States.

A caravan of hundreds of Central American migrants had trekked through Mexico a few months before, seeking passage into the United States. American officials wanted to know: Would Mexico agree to force such migrants to apply for asylum there, instead of letting them enter the United States?

The Mexicans said no.

Today, with many thousands more in transit, by far the largest single movement of migrants north in decades, the pressure to resolve the issue has reached new heights.

With a proposal broached last July known as a “safe third country agreement” seemingly off the table — it has been rejected by Mexico’s president-elect — Mexico again finds its migration system in a state of crisis, and in the cross hairs of American officials.