He also said he's prepared to slap new tariffs on China if continued trade talks are unsuccessful.
A day after Mexican officials came to the White House for talks, Trump says "a lot of progress" was made but that he's prepared to take "dramatic" steps against the country.
"I think a lot of progress was made yesterday," Trump said in Ireland before departing for Normandy.
"Something pretty dramatic could happen," Trump said. "We've told Mexico the tariffs go on. And we mean it, too."
Trump has threatened successively higher tariffs on Mexico if they don't take steps to curb illegal border crossings.
In May, more than 144,000 migrants were encountered or arrested at the US-Mexico border, a roughly 32% increase over April and the highest monthly total in 13 years, according to US Customs and Border Protection.
Of those, nearly 133,000 had crossed illegally, including more than 11,000 unaccompanied children.
In his short remarks at the Shannon airport, Trump hammered Democrats for failing to take legislative steps that would help alleviate swelling numbers of border crossings.
"They don't care what kind of record they have," Trump said. "We could solve this problem so easily if Democrats in Congress were willing to make some changes."
Trump also "a lot of interesting things are happening" in trade talks between Washington and Beijing.
"We'll see what happens," Trump said, noting the US has applied tariffs of 25% on $250 billion worth of Chinese goods.
"I can go up another at least $300 billion and I'll do that at the right time," he said. "I think China wants to make a deal badly."
Trade negotiations between the US and China stalled last month after a prospective deal fell through.
The talks between the White House and Mexico officials will continue later Thursday after no deal was reached on immigration and tariffs.
Trump tweeted Wednesday evening that he intends to move ahead with his threat to impose a 5% levy on all Mexican imported goods if no agreement is reached on immigration enforcement.
The comments came after Vice President Mike Pence had a 90-minute White House meeting with Mexico's foreign minister, Marcelo Ebrard, on Wednesday. A senior administration official told CNN the officials from Mexico put forward things they could do better and acknowledged there's a crisis on the US-Mexico border, but administration officials say it's still not enough and there's more Mexico can do.
Pence reiterated Trump's assessment that "not nearly enough progress was made" during Wednesday's meeting.
Ebrard said the talks with US officials were focused on migration, not tariffs. He acknowledged that the number of migrants has gone up, and said it will be a subject discussed in the continuing talks Thursday.
"We didn't discuss the tariffs. The dialogue was focused on migration and what Mexico is proposing to the United States," Ebrard said.
"We had the opportunity to share our point of view," he added.
It capped a roller-coaster day in Washington over the threatened tariffs, which the President has said he would increase to 25% by October if there is no change in migration across the southern US border -- a move that economic analysts say risks hundreds of thousands of US jobs.
Republicans in Congress have expressed open opposition, challenging Trump to defend his plan to them in person and making clear they want him to find a way out of the tariff threat he issued last week.
Members want him to appear before them so they can appeal directly to him and urge him to back down.
Sen. Roy Blunt, a Missouri Republican, said Trump should make his case to them in person.
"I think we have had in the past effective meetings with the President," he said.
A senior GOP official told CNN there is "no indication" the White House is any closer to an agreement that would stop Trump from imposing tariffs on Mexico.
The White House has provided little new information to Republican senators since the "scant information" offered at a meeting Tuesday between the lawmakers and White House and Justice Department officials, the official said.
And it doesn't look like Trump will meet with senators personally to hear their concerns, after Majority Leader Mitch McConnell suggested it to White House officials at the end of the meeting, according to the official.
The official said there is a "fear" that the tariffs not only would harm the economy but also would jeopardize the new United States-Mexico-Canada trade agreement, as well as new border money aimed at stemming the flow of migrants.
Factors at play
At issue is whether the President will declare a new national emergency to proceed with the escalating tariffs, which he announced last week on Twitter as a move intended to pressure Mexico into stemming Central American migrant traffic across the southern US border.
White House counsel Pat Cipollone will meet with legal counsel from Mexico on Thursday. There also will be additional, lower-level conversations Thursday, the senior administration official told CNN.
White House officials told CNN earlier on Wednesday that they were not expecting a deal to come out of the meeting between the vice president and Mexico's foreign minister. Instead, one official said, the meeting would be "an opportunity for the Mexican government to show what more they're doing" to avoid the tariff on imports.
Yet top White House officials have suggested that Trump will drop the tariffs if he gets an offer he likes from Mexico, which has already moved to beef up its enforcement activities.
Trump's top trade adviser, Peter Navarro, said Wednesday the threatened tariffs "may not have to go into effect" if the country can demonstrate it's taking steps to halt the flow of undocumented migrants across the southern US border.
CNN's Natalie Gallon, Haley Byrd, Manu Raju, Ted Barrett, Jim Acosta, Devan Cole, Eric Fiegel and Priscilla Alvarez contributed to this report.