Mr Trump has previously threatened to cut Honduran aid.
"For that reason we are soliciting the help and support of the United Nations, so that the determination of the requests for refuge will be done in transparency and with the support of the worldwide community", stated Videgaray.
He instead deployed the National Guard to the border.
Mexico has sent two Boeing 727s carrying federal forces and riot police to Tapachula, in readiness for the caravan's arrival.
In part, the response is meant to both dissuade the migrants from even attempting to enter Mexico and potentially to show the Trump administration that Mexico is willing to, as the government put it, "maintain order".
Mexico has rushed to contain the situation.
The deal comes just a few months before Mexican President Enrique Peña Nieto (pictured above right) leaves office.
Migrant caravans of this type are nothing new, they quite often take place.
Trump tweeted Thursday that he wanted "Mexico to stop this onslaught".
Why does it concern Trump?In press interviews, the immigrants are gaining widespread popularity for denouncing the inequality, corruption and state violence that plague all of Central America.
Mr Trump blamed the migrant crisis on the opposition party, the Democrats, branding its immigration policy "weak".
Mexico is seeking global aid as the country prepares for the arrival of thousands of migrants.
"It was a great sacrifice, but it's all for a better life", he said.
But foreign minister Videgaray said that was "false" and that no such deal had been made.
Where is the caravan now?
Several thousand Honduran migrants moved this week through Guatemala, and some were trying to cross to Mexico on Thursday, according to local media.
Many were travelling with a single change of clothes and little money. Mexican officials also said migrants caught without papers would be deported.
Thousands of people and goods legally cross the US-Mexico border daily.
Arpaio spoke as President Trump on Thursday vowed to potentially utilize the USA military to "close our Southern border". Only about 150 made it to the border crossing at Tijuana, seeking asylum.
The last time a similar caravan of immigrants sought to cross into Mexico, in April, authorities gave its participants short-term visas that allowed them to travel through Mexico and reach the United States border. Trump said in a Wednesday morning tweet.
But the mass exodus also drew the ire of Trump, who warned that governments in the region that allow this kind of migration face a possible end to USA aid, putting considerable pressure on even US allies.
Irineo Mujica, the director of the organisation Pueblo Sin Fronteras, was detained at a march in Ciudad Hidalgo, southern Mexico that was supporting a new group of around 4,000 migrants.
One of the migrants, Juan Escobar, 24, said he had heard about Trump's comments but said they would not dissuade the migrants from continuing their journey.
Human rights groups have criticised the United States and Mexican response to the caravan.
In a statement, the country's Institute of Migration (IGM) said: "Guatemala does not promote or endorse irregular migration in any forms, and therefore rejects movements organized for unlawful purposes which distort human rights, like migration, for their own end". López Obrador, the president-elect, will take over December 1.
That's a tiny portion of the nearly 1,350 miles (2,170 kilometers) they'd have to travel to reach the closest USA border.