One idea that Washington has proposed for years is a “safe third country” arrangement in which migrants who flee persecution in Central American countries would first have to apply for asylum in Mexico. Those who do not could be turned away if they seek refuge in the United States. Mexico has long opposed the idea.

But there appeared to be a significant disagreement on Monday between the Mexican government and American officials about what the negotiators actually agreed to regarding further action and the possibility of implementing a “safe third country” arrangement.

Administration officials characterized the Mexicans as having all but agreed to asylum changes that would effectively mimic the benefits of “safe third country” as long as it was done regionally, including countries like Guatemala.

Mr. Trump did not specifically mention the idea, but said Monday that “we have an agreement on something they will announce very soon. It’s all done.”

But that was not the understanding described by Mr. Ebrard at his Monday news conference. He told reporters only that Mexican officials would discuss changing asylum rules if the flow of migrants was not substantially reduced in the next several months.

“They will propose safe third country,” Mr. Ebrard said, describing what he expects would be the United States position in any future discussions. “We said it will have to be with the U.N.H.C.R., it will have to be regional,” he said, referring to the United Nations High Commissioner for Refugees.

Mr. Ebrard said Mexico preferred a regional asylum agreement that would review the flow of migrants across Mexico and Central America, with a number of countries, including Panama and Brazil.