Mexico will also reinforce its southern border with Guatemala with thousands of National Guard officers.
The Times, however, said that sources noted this pledge was made in March during talks between then-Department of Homeland Security Secretary Kirstjen Nielsen and Mexican Secretary of the Interior Olga Sanchez.
The centerpiece of Mr. Trump's deal was an expansion of a program to allow asylum-seekers to remain in Mexico while their legal cases proceed. But that arrangement was first reached in December in a pair of painstakingly negotiated diplomatic notes that the two countries exchanged. Now, he said, "I have full confidence, especially after speaking to their President yesterday, that they will be very cooperative and want to get the job properly done".
After nine days of uncertainty, Mr. Trump backed down and accepted Mexico's promises.
A former director-general of the World Trade Organization (WTO), Pascal Lamy, said it was understandable that Mexico had sought to extricate itself from the tariff bind, but said Mexico ran the risk of more threats from Trump in the future.
The Mexican government is promising humanitarian aid to Central American refugees blocked in the country because of new tighter controls on the United States border.
This image from January shows a migrant from Honduras holding a sign that says "no to the wall" in Reynosa, Mexico.
House Speaker Nancy Pelosi said Trump had "undermined America's pre-eminent leadership role in the world by recklessly threatening to impose tariffs on our close friend and neighbor to the south".
Mexican President Andrés Manuel López Obrador tweeted, "Thanks to the support of all Mexicans, the imposition of tariffs on Mexican products exported to the U.S. has been avoided".
The talks had been focused, in part, on attempting to reach a compromise on changes that would make it harder for migrants who pass through Mexico from other countries to claim asylum in the U.S., those monitoring the situation said. "We should not be confronting them every other day". To date, migrants have been returned at only three of the busiest ports of entry. What kept that from happening, they said, was the commitment of resources by both countries.
"Mexico, in turn, has agreed to take strong measures to stem the tide of Migration through Mexico, and to our Southern Border". But the speed with which the United States can send more migrants to wait in Mexico will still depend on how quickly the government follows through on that promise. "We have an agreement that, if they implement, will be effective".
After the last night announcement of the Mexico deal, Sen.
"We need a decent relationship with Mexico", added Sanders, a candidate for his party's presidential nomination in 2020.