MEXICO CITY (REUTERS) - Mexico avoided the most extreme immigration concession sought by United States President Donald Trump in the deal reached to fend off threatened tariffs.

But it is left even weaker than before in the face of potential new pressure from Mr Trump as he formally kicks off his re-election campaign this month.

Under the deal reached last Friday (June 7), Mexico agreed to use a large part of its newly formed National Guard to hold back immigrants crossing from Guatemala, and to take in possibly tens of thousands of people seeking asylum in the US while their cases are adjudicated.

Mr said on Sunday his administration will release additional details on its agreement with Mexico as he denied a news report that there were no new major commitments made by Mexico to stem the flow of Central American migrants at the southern border.

"Mexico was not being cooperative on the Border in things we had, or didn't have, and now 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," Mr Trump said on Twitter.

"Importantly, some things not mentioned in yesterday press release, one in particular, were agreed upon. That will be announced at the appropriate time," he added.

Led by Foreign Minister Marcel Ebrard, negotiators in Washington resisted Mr 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 US.

But the two sides agree more action could be taken if the measures do not have the desired result within 90 days, of drastically bringing down the number of undocumented migrants reaching the US border from levels that are at a more-than-decade high. Last month alone, 132,000 were apprehended by US authorities.

Former World Trade Organisation head Pascal Lamy called Mr Trump's approach to coercing its neighbour and ally "hostage-taking", reflecting widely held concerns in Mexico that the US President would come back with more threats to extract greater concessions.

Those fears were sharpened because Mr Trump had used Mexico-bashing to fire up his base on repeated occasions since kicking off his first campaign for the White House in 2015. All signs indicate that he intends to keep the focus on immigration and cross-border issues in his second-term campaign, which officially launches on June 18.

"We think the threats, demands and Trump tweets against Mexico will continue, especially because it's all tied up with the politics of the 2020 election," said Ms Gabriela Siller, an economist at Mexican bank Banco Base.

Ms Siller expects the peso currency to rise when markets open on Monday on relief that a trade war has been averted, but she said the uptick could be short lived. The peso, which had been pummelled in recent months on fears over a trade war, strengthened 0.5 per cent last Friday after Mr Trump tweeted that there was a "good chance" a deal would be reached with Mexico.

Mr Vicente Fox, a former Mexican president and long-term critic of President Andres Manuel Lopez Obrador, tweeted that by allowing the US to dictate how, for example, Mexico uses its security forces, the government has already ceded some of its sovereignty.

That sentiment was echoed by centre-left politician Angel Avila, on the other side of the political aisle from Mr Fox, who called the deal "a surrender".

"Mexico shouldn't militarise its southern border," said Ms Avila, who heads the Party of the Democratic Revolution.

Others, however, think that President Lopez Obrador had little choice beyond giving some ground in the negotiations, because the threatened tariffs would have caused economic devastation in Mexico, whose economy contracted in the first quarter of this year.

Mr Francisco Labastida, a former presidential candidate, said the scale of the current immigration crisis was a threat to Mexico itself, and action was needed regardless of Mr Trump.

"Mexico would have had to change its migration policy for its own reasons due to national security," he said, arguing that current numbers are unmanageable.

Mr Carlos Pascual, a former US ambassador to Mexico, praised the deal as preferable to the downward spiral of a tit-for-tat trade war, but acknowledged it left Mexico open to further pressure.

"Mexico is weak economically and it's always going to be vulnerable if the United States is willing to use economic policy to enforce national security policy," he said.

"There's no doubt this leaves a Damocles sword hanging over Lopez Obrador's head," he said, invoking a metaphor that describes an ever-present peril.