Updated at 9:58 a.m. ET

New tariffs against Mexico will begin to bite next week, President Trump vowed Tuesday, unless the White House is satisfied that Mexico's government is acting with new alacrity to stop migrants from crossing into the United States.

"This will take effect next week, 5 percent," Trump said.

Trump said he's open to continuing negotiations with Mexican leaders, including at a meeting scheduled for Wednesday between its foreign minister and Secretary of State Mike Pompeo.

The president is also willing, he said, to impose the tariffs during the ongoing negotiations to make clear he is serious.

"I think Mexico will step up and do what they should have done," Trump said at a press conference with British Prime Minister Theresa May after a meeting with her at 10 Downing Street.

Trump said he won't accept any explanation about the difficulty for Mexican authorities of constraining the flow of migrants from Central and South America.

"I don't want to hear that Mexico is run by cartels and drug lords and coyotes," he said. "I don't want to hear about that ... Mexico should step up and stop this onslaught, this invasion into our country."

Trump at Number 10

Trump and May also concluded a series of meetings with general commitments to pursue a new trade deal — although apparently without one in place. That has taken on new urgency for May and Britons, as they eye the United Kingdom's pending exit from the European Union, which has jangled the British economy and its politics.

That is why May is nearing the end of her premiership and a leadership contest is underway within her Conservative Party to replace her.

Trump hasn't been shy before and during his visit about expressing his thought on a successor. He spoke by phone with former Foreign Minister and London mayor Boris Johnson and said on Tuesday he thought Johnson "would do a good job."

The leader of the Labor Party, meanwhile, Jeremy Corbyn, wanted to meet with Trump, but the president declined, Trump said. It wasn't immediately clear whether that was correct.

Corbyn has faulted Trump's politics, and he was said to be addressing anti-Trump protesters in Central London around the time of the press conference between the U.S. and British leaders.

Copyright 2019 NPR. To see more, visit https://www.npr.org.