In a statement, the ministry said that the US government must determine whether it would allow any of the migrants from the caravan into the United States.
"It is not up to this government to make migration decisions for the United States or for any other nation," the ministry said.
The ministry also said it did not foster illegal migration "under any circumstances".
It comes after Trump took to Twitter to attack Mexico for allowing the 1,500-strong caravan to march unimpeded toward the United States, their daily progress mapped out by US media.
He threatened to abrogate NAFTA, the US-Mexico-Canada free trade pact, and demanded Congress pass tougher immigration legislation and support his plan for a wall along the Mexican frontier.
He also declared he no longer supported a replacement for DACA, the program that had offered protection against deportation for about 700,000 undocumented immigrants, most of them Mexicans, who came to the US as children.
"Mexico has the absolute power not to let these large 'Caravans' of people enter their country," the president wrote.
"Mexico is making a fortune on NAFTA... They have very strong border laws -- ours are pathetic," he added.
"With all of the money they make from the US, hopefully they will stop people from coming through their country and into ours, at least until Congress changes our immigration laws!"
Since taking office more than a year ago, cracking down on legal and undocumented immigration has been a focus of Trump.
He has repeatedly tried to link immigration with crime but immigrant defenders say many are fleeing poverty and violence.
1,500 headed to border
Mexico's Interior Minister Alfonso Navarrete Prida rejected Trump's accusations.
"It is absolutely incorrect to say that Mexico is not making an effort" to control illegal migration, he told journalists on Monday.
"We have been following the course of this caravan, whose numbers have fallen significantly.... Of course we will act, let me be clear, in strict compliance with our migration laws, without accepting pressure from any country or anyone."
Organised each of the past five years by activists of Pueblo Sin Fronteras, or People Without Borders, the 1,500 migrants have fled Honduras, Guatemala, Nicaragua and El Salvador because of poverty and widespread violence from criminal gangs, they say.
They set off on March 25 from the southern Mexican state of Chiapas for the US border, where many hope to apply for asylum.
On Monday they were passing through the mountainous southern Mexican state of Oaxaca.
Trump appears to have become aware of them over the weekend from a report on Fox News, which he regularly watches.
"Mexico is doing very little, if not NOTHING, at stopping people from flowing into Mexico through their Southern Border, and then into the US. They laugh at our dumb immigration laws. They must stop the big drug and people flows, or I will stop their cash cow, NAFTA," he tweeted on Sunday.
He demanded Congress pass tougher immigration laws and give more support to building a wall along the border. Accusing the migrants of seeking to benefit from DACA, he declared no deal would be made to replace it.
A leader of the caravan countered that Trump is using it as a "pretext" to stop any deal on DACA that would provide those qualified -- as many as 1.8 million people -- a path to citizenship.
"This is just a pretense to lay the blame somewhere else. He's just using our movement, the caravan, as a pretext," caravan organiser Rodrigo Abeja told AFP.
Abeja said the Mexican government was in a difficult position with the caravan, given the growing publicity attached to its daily progress.
"The Mexican government is under a lot of pressure from the United States to show that it has control over its borders. But it also can't take direct action (such as deporting the migrants) because there is already a lot of public exposure," he said.
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