Trump also said that if a solution is not found, it could upend the new trade deal negotiated between Mexico and the United States.
Trump shared the footage on Twitter, praising the response.
Trump has made cracking down on illegal immigration and building a wall that stretches the length of the 1,954-mile (3,145 km) border a keystone of his presidency, but his animus toward Mexico had cooled since Obrador's election in July.
Hundreds of Honduran migrants stand at the shore of the Suchiate river on the border between Guatemala and Mexico, in Tecun Uman, Guatemala, Oct. 18, 2018.
"Republicans are fear-mongering about a migrant caravan to boost midterm turnout" said a Salon headline, while CNN noted, "Mother in caravan: Trump has to receive us".
Vigil said he believes Trump is partly to blame for the recent exodus from Honduras because of US involvement in the country's contested presidential election past year.
The Mexican consulate in Guatemala published a statement saying immigrants who wish to enter Mexico must have "travel documents and a visa" and that those who "enter into the country in an irregular manner" will be returned "to their country of origin, in a secure and orderly manner".
He also threatened to mobilize the US military and shut down the southern border to stop those migrants from entering the USA, reports CBS News correspondent Chip Reid.
Not enough progress has been made on Brexit, EU leaders say
However, according to the European Parliament president Antonio Tajani, who was also in the room, May offered no new ideas. CBC's Margaret Evans looks at what could potentially happen as Britain barrels towards Brexit.
Trump's frustration appears to stem in part from record levels of immigrant parents entering the USA with children in the three months since his administration ended family separations at the border.
The row comes as US Secretary of State Mike Pompeo prepares to embark on a tour of the region that will see him visit Mexico ahead of its December inauguration of President-elect Andres Manuel Lopez Obrador, as well as Panama.
"They wanted that caravan and there are those that say that caravan didn't just happen", Mr. Trump said. Immigration officials later issued a statement levying property damage allegations against Mujica, who holds dual U.S. He was referring to the United States-Mexico-Canada Agreement, which is awaiting ratification.
Gutierrez told Fox News that his country would accept those with genuine asylum claims.
U.S. Representative Michael McCaul, the Republican chairman of the House Homeland Security Committee and a member of the House Foreign Affairs Committee, said U.S. border security was a key election issue. The foreign minister, Luis Videgaray, asked the United Nations to assist with what could be a crush of requests.
The caravan has been growing steadily since it left the violent Honduran city of San Pedro Sula on Saturday.
"The electoral process is very near, so he is making a political calculation", Ebrard said in an interview with Radio Centro.