Asked in the Televisa interview whether Mexico was doing Trump's "dirty work", Videgaray said Mexico "defines its migration policy in a sovereign manner" and the country's priority is to protect the migrants and ensure their human rights.
Mexican authorities have met and spoken with the migrants, telling them 100 would be allowed through each day to be processed.
Some then rushed a border fence and attempted to push their way through, prompting police to deploy smoke canisters, according to the AP.
"In Mexico we have the rule of law, and we shall apply, enforce the law", Videgaray told reporters.
Hundreds of Mexican police were sent to guard the border between the Guatemalan town of Tecun Uman and Ciudad Hidalgo in Mexico to prepare for the migrant caravan's arrival. "We don't want them", Trump said.
On Thursday, Videgaray asked the United Nations for help processing what Mexico expects to be a large number of asylum requests.
"We are quickly reaching a point, which appears to be a moment of crisis, record numbers of migrants", added U.S. Secretary of State Mike Pompeo.
Mexican Foreign Relations Secretary Luis Videgaray said those with passports and valid visas would be let in immediately, though he acknowledged that "we anticipate those are the minority". It was unclear if they were trying to swim across to the Mexico side of border, go back to Guatemala or just get relief from the crowds on the bridge.
Police and immigration agents began letting small groups of 10, 20 or 30 people through the gates if they wanted to apply for refugee status.
Free buses started arriving late on Friday night, just hours after Mexican riot police had hurled teargas bombs into the frustrated crowd after some migrants broke through a riot-shield cordon to enter Mexican territory. (JOHAN ORDONEZ/AFP/Getty Images)What have officials been saying?
On Friday afternoon, hundreds of the migrants poured through Guatemala's frontier posts towards the closed Mexican border on a bridge spanning the Suchiate River that bisects the two countries, Mexican television footage showed.
Witnesses say some of the migrants threw rocks and other objects at Mexican security forces, who used pepper spray to force the migrants to retreat.
"Violent entry into the country not only threatens our sovereignty, but also puts the migrants themselves at risk", Pena Nieto said in a video published on his social media profiles.
A chaotic scene unfolded as a massive caravan of Honduran migrants reached the Guatemala-Mexico border Friday.
Hernandez and Morales are expected to meet in Guatemala early Saturday to discuss the situation.
NBC reports that United States officials are now tracking that small group as they make progress towards the US. "It's going to be an election of the caravan", Mr Trump said at a campaign rally in Missoula, Montana. God is going to soften Trump's hard heart.
The president has repeatedly claimed that the migrants, and immigrants more broadly, lead to increased crime rates and allow illegal substances to pass over the border, although statistics suggest they do not. We're calling up the military.
"Only God on high can stop us", Escobar said. They'd been traveling since 4 a.m. and arrived at the border after dark. They might as well turn back. "That would really hurt legal transit between the two countries and manufacturing and trade, which would affect American workers", Selee said.