Earlier in the morning, Trump made headlines for his Twitter tirade against immigrants.
Already some migrants from the caravan have begun to arrive, seeking refuge, Mexico's foreign ministry said in a statement Thursday night. Some hope to eventually enter the United States to escape violence and poverty.
Mexican authorities, in search of a way to satisfy President Trump's demands to deter the migrants without appearing to violate worldwide law, have asked the United Nations to set up a migrant processing center near its southern border.
This includes a visit to Mexico ahead of its December inauguration of President-elect Andres Manuel Lopez Obrador. "It's the third time that I'm trying to cross", the 22-year-old Salvadoran said. "It's very important to have a relationship of friendship".
They got as far as a closed metal gate where two military jeeps were parked and Guatemalan police in riot gear looked on silently.
But experts on refugees say those moves are unlikely to stop this caravan or stem future ones, since the migrants are primarily driven by conditions in their home countries, not USA policy. "The United States stands ready to assist the Government of Mexico and UNHCR in this effort".
Thousands of people and goods legally cross the US-Mexico border daily.
"One way or another, the way Mexico treats the undocumented immigrants from Central America affects Mexico's position in how it can criticize the US government for its treatment of undocumented Mexican immigrants in the United States", Salazar told CNN en Español.
In 2016, it was estimated that 400,000 migrants pass through Mexico every year. The nearest border is about 110 miles (177 km) away.
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"We are not leaving because we want to", he said. The US no longer has to house them until their cases are heard.
Under pressure from the United States to stop the migrants, Mexico has sent federal police to the border, though they are not officially there to stop the caravan.
With less than three weeks before the November 6 midterm elections, Trump seized on the migrant caravan to make border security a political issue and energise his Republican base.
Central American governments have dismissed Trump's threats to block United States aid.
Trump tweeted that he wanted "Mexico to stop this onslaught".
The map above shows the journey the caravan is taking from Honduras through Guatemala and up through Mexico into the U.S. "Great Midterm issue for Republicans!".
Gutierrez told Fox News that his country would accept those with genuine asylum claims.
The countdown is on for the arrival of what has come to be known as the "Honduran caravan" - an estimated 4,000 men, women, children and infants who left the troubled Honduran city of Pedro Sula last weekend on foot and by truck. There are no official estimates of the size of the group.
Guatemalan President Jimmy Morales said on Wednesday his government dismissed threatened constraints placed on foreign aid.
Fulton also repeated Trump's earlier threat to detain and deport anyone entering the US illegally, a policy the administration has referred to as "zero tolerance".