Trump reacted by threatening to end the North American Free Trade Agreement (NAFTA) and also end foreign aid to Honduras, where numerous migrants originated, if the caravan was not stopped.
Buzzfeed News, which first reported the caravan's movements last week, reported on Tuesday that organizers said the caravan would travel to Mexico City, where only those with valid asylum claims in Mexico or the US would continue moving north to the border.
"President Trump: If you wish to reach agreements with Mexico, we stand ready, as we have proved until now, always willing to engage in a dialogue, acting in earnestness, in good faith and in a constructive spirit", Peña Nieto said, according to the Washington Post.
The Pentagon could not say where the money would come from and Trump admitted the White House was still "looking at" costs.
Pena Nieto has faced criticism for failing to take a tough line against Trump in the past.
Trump's series of comments this week directed at Mexico have thrust the countries' relationship into the center of the presidential campaign, where the ruling party is trailing.
"This great threat on the southern border of the United States that he says is there, does not exist", said the leftist Lopez Obrador, who has a double-digit lead in most polls.
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Andres Manuel Lopez Obrador, the sometimes fiery leftist gunning to be Mexico's next president, disputed Trump's anti-migration arguments, saying that crime is actually down in USA border cities, that less Mexicans are crossing the border and that deportations have fallen under Trump.
"This anti-Mexican policy has worked politically because unfortunately there are conservative sectors in the United States with little information and he knows how to awaken an anti-Mexican sentiment, what's called xenophobia, which can be like racism, hatred of foreigners", Lopez Obrador said.
On Tuesday, the Mexican government asked the United States for clarification on the menaces made by Trump to send the armed forces to the border while the wall that separates his country from Mexico is being lifted.
"You can not negotiate or cooperate with threats", he said.
Barring Meade, all of the candidates are trenchant critics of the president, but Pena Nieto namechecked all four of them and the Senate motion in his scolding of Trump.
The move, about which specifics were unclear, appeared to be prompted in part by news reports of the caravan moving toward the U.S. -Mexico border.