"There is a clear limit to what we can negotiate, and the limit is Mexican dignity", said Mexico's Ambassador to the United States, Martha Barcena. "Everyone's coming through Mexico including drugs, including human trafficking...we're going to stop it or we're not going to do business, and that's going to be it".
If a deal is not reached, however, the president said his plan will be enforced. More than a quarter million more migrants would reach the USA this year from Central America without Mexico's help, he added.
If Congress did not approve the United States-Mexico-Canada Agreement - Trump's proposed successor to the 1994 North American Free Trade Agreement - he could face an embarrassing setback ahead of the 2020 elections.
The talks in Washington will be closely watched by financial markets concerned that import tariffs would ultimately hit the United States economy by adding to the cost of a wide range of goods in the USA, from Mexican-made cars and auto parts to beer and avocados.
Trump said on Thursday that he would apply the tariffs on June 10 if Mexico did not halt the flow of undocumented migration, largely from Central America, across the US-Mexican border.
The discussions in Washington will include a meeting of Mexican Agriculture Minister Victor Villalobos and US Agriculture Secretary Sonny Perdue on Monday, Mexican officials said. "I don't think the president's going to impose these tariffs". Those tariffs are set to increase to 10% on July 1 and will continue increasing by 5% until Mexico addresses the immigration matter. Many seek asylum in the U.S. when they cross the border.
The BBC says Mexico was the second largest supplier of goods to the United States a year ago, with imports totaling $352bn, according to Goldman Sachs.
The president "has been known to play with fire, but not live hand grenades", Kennedy said on CBS's Face the Nation.
Mexican officials in Washington warned on Monday that the tariffs could backfire, fanning further migration by hammering regional economies.
Cornyn expressed some hope that Trump still could be persuaded to change course on his latest tariff threat.
Trump claims Mexico has taken advantage of the USA for decades but that the abuse will end when he slaps tariffs on Mexican imports next week in a dispute over illegal immigration.
"I'm certainly very uneasy about it".
Sen. Joni Ernst (R-Iowa) told reporters on Monday that she had spoken to Trump about his tariff threat over the weekend and urged him to back down, but he appeared to be unmoved.
Apprehensions at the USA border with Mexico have surged in recent months, though Mexican data also show more deportations and detentions at Mexico's southern border with Guatemala, mostly of Central Americans trying to reach the United States. "The Tariff will gradually increase until the Illegal Immigration problem is remedied", Trump wrote.
The tariffs are threatening to disrupt supply chains for many USA companies.
The tariff threat was immediately met with a backlash from Mexico, businesses, and politicians from both sides of the aisle.