In a pair of tweets Monday morning, Trump claimed that Mexico had agreed to more than what was revealed in the Friday announcement, teasing that more would be announced soon.

STERLING, Va. — Stung by criticism that his deal to avert threatened Mexican tariffs mostly ramps up existing efforts, President Donald Trump is insisting there's more to it than meets the eye.

In a pair of tweets Monday morning, Trump claimed that Mexico had agreed to more than what was revealed in the Friday announcement, teasing that more would be announced soon.

"We have fully signed and documented another very important part of the Immigration and Security deal with Mexico, one that the U.S. has been asking about getting for many years," he wrote, claiming that it would be "revealed in the not too distant future and will need a vote by Mexico's Legislative body!.."

"We do not anticipate a problem with the vote," he added, "but, if for any reason the approval is not forthcoming, Tariffs will be reinstated!"

White House officials did not immediately respond to requests for comment about what Trump was referencing. And Trump would not say during a call-in interview with CNBC Monday morning. But he could be alluding to the idea of Mexico becoming a "safe third country," which would make it harder for asylum-seekers who pass through the country from other places to claim refuge in the U.S.

A senior administration official said over the weekend that Mexico had expressed openness to the idea during negotiations, and that it was something the countries would continue to discuss over the coming months. The official spoke on condition of anonymity to share details of closed-door talks.

Mexico, however, has long opposed the safe country idea and Mexican Foreign Minister Marcelo Ebrard said on Friday after a deal was reached that Mexico had resisted. The U.S. "proposed in the first meeting that we have a third safe state, which is not the case, which is very important," he told reporters.

The tweets came amid questions about just how much of the deal — announced with great fanfare Friday — was really new.

It included a commitment from Mexico, for instance, to deploy its new National Guard to the country's southern border with Guatemala. Mexico, however, had already intended to do that before Trump's latest threat and had made that clear to U.S. officials. Mexican officials have described their commitment as an accelerated deployment.

The U.S. also hailed Mexico's agreement to embrace the expansion of a program implemented earlier this year under which some asylum-seekers are returned to Mexico as they wait out their cases. But U.S. officials had already been working to expand the program, which has already led to the return of about 10,000 to Mexico, without Mexico's public embrace.

"The president has completely overblown what he reports to have achieved. These are agreements that Mexico had already made, in some cases months ago," said Democratic presidential candidate Beto O'Rourke, speaking on ABC's "This Week." ''They might have accelerated the time table, but by and large the president achieved nothing except to jeopardize the most important trading relationship that the United States of America has."

Trump has pushed back on that criticism, defending the deal and his threat to slap a 5% tax on all Mexican goods on Monday to pressure the country to do more to stem the flow of Central American migrants across the U.S. southern border. Without the threat, he has insisted, Mexico never would have acted.

"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 tweeted Sunday, adding during a call-in interview with CNBC Monday morning that officials had "talked about it for months and months and months," but couldn't reach agreement until the threat.

Acting Homeland Security Secretary Kevin McAleenan, speaking on "Fox News Sunday" also insisted "all of it is new," including the agreement to dispatch around 6,000 National Guard troops — a move Mexico has described as an "acceleration."

"This is the first time we've heard anything like this kind of number of law enforcement being deployed in Mexico to address migration, not just at the southern border but also on the transportation routes to the northern border and in coordinated patrols in key areas along our southwest border," he said, adding that "people can disagree with the tactics" but that "Mexico came to the table with real proposals" that will be effective, if implemented.

Trump has also dangled the prospect of renewing his tariff threat if the U.S. ally doesn't cooperate to his liking.

"There is now going to be great cooperation between Mexico & the USA, something that didn't exist for decades," he tweeted Sunday.

"However," he added, "if for some unknown reason" that doesn't happen, "we can always go back to our previous, very profitable, position of Tariffs." Business leaders and many Republicans had urged Trump against the tariffs, warning they would drive up consumer prices, hinder the economy and compromise the ratification of an updated North American trade deal.

Mexico's ambassador in Washington, meanwhile, said her country was committed to cooperating with the U.S. and that discussions would continue over the coming 90 days.

"We want to continue to work with the U.S. very closely on the different challenges that we have together. And one urgent one at this moment is immigration," said Martha Barcena. She told CBS' "Face the Nation" that the countries' "joint declaration of principles ... gives us the base for the road map that we have to follow in the incoming months on immigration and cooperation on asylum issues and development in Central America."

The U.S., she said, wants to see the number of migrants crossing the border return to levels seen in 2018. U.S. Border Patrol last week announced it had apprehended more than 132,000 people at the border in May, including a record 84,542 adults and children traveling together, straining federal resources and leaving officials struggling to provide basic housing and health care.