"We are global workers!" "All will stay in Mexico", the president wrote.
He said migrants would not be allowed entry until their claims were approved.
Migrants approaching the US border from Mexico have been enveloped with tear gas after a few tried to breach the fence separating the two countries.
Without confirming the deal, James McCament, Homeland Security's Acting Under Secretary for Policy, said the USA government has been working since the Mexican elections with its current and incoming Mexican counterparts on trade, border policy and other issues.
Sanchez, who said the situation of migrant caravans was "very delicate", did not explicitly rule out that Mexico could keep caravan migrants on its soil while their U.S. asylum claims are processed.
However, Olga Sanchez Cordero, senator from the Morena party and future Interior Minister (takes office on December 1) in the government of Mexican President Andres Manuel Lopez Obrador, has denied this media report. "No "Releasing" into the US", Trump said in a tweet late Saturday.
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Most of the migrants are camped inside a sports complex, where they face long wait times for food and bathrooms.
Several thousand migrants, most from Central America, have been gathering in Tijuana in hope of entering the US.
The mayor of Tijuana has declared a humanitarian crisis in his border city and says that he has asked the United Nations for aid to deal with the approximately 5000 Central American migrants who have arrived in the city.
"When they come back, we need to see how ... we can integrate them into an economic activity so that they can develop and not generate conflict with our own communities".
Alison Leal Parker, US managing director for Human Rights Watch, a New York-based rights organisation, said the plan was "a pathetic attempt by the United States to shirk responsibility".
In 2018, border patrols registered more than 400,000 illegal border crossers, according to Homeland Security, and in the last five years, the number of those requesting asylum has increased by 2,000 percent.
Although the press, and in particular CNN, wish to treat this as a concotion of the administration, it seems the thousands of hopeful immigrants in the caravans, and the organizations supporting them, also take the situation very seriously and are taking action on that seriousness.
Rene Vazquez, 60, a Tijuana resident who was volunteering at the stadium, said Mexico's federal government ignored the problem by allowing the caravan to cross the country without stopping.