Republicans and Democrats warned President Trump on Tuesday to quickly strike a border deal with Mexico and drop his threat of economically crippling tariffs, saying otherwise he faces an embarrassing vote to override his move in Congress.

Democrats were more enthusiastic about the battle, but even Republicans indicated that Mr. Trump would not have their support if he follows through on his threat to slap a 5% tariff on all Mexican goods Monday.

“We’re not fans of tariffs,” said Senate Majority Leader Mitch McConnell, Kentucky Republican.

Mr. Trump, traveling in London, told Republicans not to go wobbly on him.

“I don’t think they will do that. I think if they do, it’s foolish,” the president said during a press conference with British Prime Minister Theresa May.

He said he is committed to putting the tariffs into effect and will ratchet up the level monthly until it reaches 25%.

Mr. Trump views the tariffs as leverage against the Mexican economy. He said the southern neighbor has abused good wishes for too long by allowing a massive flow of migrants to cross its territory and reach the U.S. border.

Mexico dispatched top officials to Washington this week to plead for the president to relent.

Mr. Trump said he is open to talks but seemed to doubt they would succeed before Monday.

“I think it’s more likely that the tariffs go on,” he said.

Congressional Republicans met behind closed doors with administration officials to let them know Mr. Trump won’t have their support.

Mr. McConnell wouldn’t speculate on what steps Congress will take but said he hopes Mexico agrees to Mr. Trump’s unspecified terms.

Senate Minority Leader Charles E. Schumer, New York Democrat, said if Mr. Trump does impose the tariffs, a vote will be held to rescind them.

“I believe there would be a disapproval resolution,” Mr. Schumer said.

He said he doubted the threat will reach that point because the president has a record of talking tough and then caving — particularly when his own party deserts him.

“I have a feeling that this one just popped into his head,” Mr. Schumer said. “I believe that he will back off when faced with the opposition among business, among his own Republicans.”

Mr. Trump took to Twitter on Tuesday evening to denounce Mr. Schumer as a “creep” for undermining his negotiating tactics from partisan spite.

“Can you imagine Cryin’ Chuck Schumer saying out loud, for all to hear, that I am bluffing with respect to putting Tariffs on Mexico. What a Creep. He would rather have our Country fail with drugs & Immigration than give Republicans a win. But he gave Mexico bad advice, no bluff!” he tweeted.

Mr. Trump has survived several disapproval votes, including one on his border wall emergency this year. In that instance, Congress voted to block his move, he vetoed that bill, and Republicans sustained him.

It’s not clear whether he would have the votes this time.

Senate Republicans seemed ready to challenge Mr. Trump, though gauging House Republicans’ mood is far tougher.

House Minority Leader Kevin McCarthy, California Republican, declined to handicap a possible vote but said he hopes it doesn’t get to that point.

“We have until June 10th,” Mr. McCarthy said. “I think the best outcome of this is to have Mexico assist in the matter of what’s happening on the southern border.”

Republican lawmakers seemed particularly weary of Mr. Trump’s penchant for tariffs.

Last month, they cheered when the president canceled his tariffs on steel and aluminum imports from Mexico and Canada, clearing the way for action on the trade deal that Mr. Trump’s team negotiated with those countries.

The president, though, has always been more sympathetic to trade barriers. He says they create a win-win situation: Either he gets his way or companies will be punished for doing business abroad and will move operations to the U.S., creating more jobs.

Mr. Trump, from London, dared Republicans to cross him. He pointed to his high approval ratings with Republican voters.

“I have a 90% — 94% approval rating, as of this morning, in the Republican Party. That’s an all-time record. Can you believe that?” he said.

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