Mexico has also agreed to take "unprecedented steps to increase enforcement to curb irregular migration", including the deployment of the Mexican National Guard throughout the country, especially on its southern border with Guatemala.
Under the deal, Mexico agreed to the immediate expansion along the entire border of a programme under which the United States returns asylum-seeking migrants to Mexico to await adjudication of their cases.
US Homeland Security officials have been ramping up slowly, and were already working to spread the program along the border before the latest blowup.
"The president has completely overblown what he reports to have achieved".
"I did not contradict the president of the U.S. @realDonaldTrump", Ms. Barcena tweeted.
"These are agreements that Mexico had already made, in some cases months ago", former congressman Beto O'Rourke said in an interview on ABC.
Another 2020 candidate, Sen. "You are making a Giant Mistake!"
"I think what the world is exhausted of, and what I am exhausted of, is a president who consistently goes to war, verbal war, with our allies", Sanders said, adding that Trump should focus on achieving comprehensive immigration reform.
"I think the biggest message here is not to Mexico but to China - that the president is clearly willing to use tariffs, and actually the president believes that tariffs are a significant positive economic tool", he said, adding that he doesn't agree.
Trump echoed the same in his tweets, insisting the deal was being misrepresented.
"We have been trying to get some of these Border Actions for a long time, as have other administrations, but were not able to get them, or get them in full, until our signed agreement with Mexico", he wrote.
Trump then hinted that there was at least one surprise in the agreement, one that had not been announced in the initial press release and would be made public "at the appropriate time".
Led by Foreign Minister Marcel Ebrard, negotiators in Washington resisted Trump's core demand that Mexico be declared a safe third country, a classification that would oblige Central Americans crossing through Mexico to seek safe haven there, not the United States.
Leaving the State Department Friday night, Mexican Foreign Secretary Marcelo Ebrard said he thought the deal struck "a fair balance" because the USA "had more drastic proposals and measures at the start".
His aides said yes, but admitted that they were also realistic that the surge of immigration might continue.
But there remains deep skepticism among some American officials - and even Mr. Trump himself - about whether the Mexicans have agreed to do enough, whether they will follow through on their promises, and whether, even if they do, that will reduce the flow of migrants at the southwestern border.