President Trump said Thursday he was giving Mexico and Canada a chance to renegotiate the North American Free Trade Agreement, but warned them that he was ready to pull out if it wasn’t rewritten as a winning deal for the U.S.

In backing away from the NAFTA exit, Mr. Trump nevertheless checked off a top item on his 100-day agenda by forcing Mexico and Canada to the bargaining table over the three-country trade deal that has been the bane of the president’s blue-collar base.

The maneuver even won begrudging support from Mr. Trump’s foes within the Republican establishment, who admitted the decades-old free trade zone needed to be updated.

“I decided rather than terminating NAFTA, which would be a pretty big, you know, shock to the system, we will renegotiate,” said Mr. Trump. “If I’m unable to make a fair deal for the United States, meaning a fair deal for our workers and our companies, I will terminate NAFTA. But we’re going to give renegotiation a good, strong shot.”

The administration had been preparing to quit the deal using an escape clause known as Article 2205. That prompted Mexican President Enrique Pena Nieto and Canadian Prime Minister Justin Trudeau to call Mr. Trump and ask instead for a NAFTA redo.

“We have to make a deal that is fair for the United States. They understand that,” said Mr. Trump in the Oval Office while meeting with Argentine President Mauricio Macri.

Mr. Trump stressed that he was still willing to quit the deal.

He later added that the renegotiations were already underway. “It’ll start very soon. It’s actually starting today,” he said.

Fixing NAFTA or getting the U.S. out of it was a prominent campaign promise for Mr. Trump. It topped his 100-day agenda for action to protect American workers that he promoted during the race.

Since taking effect in 1994, the free trade zone created by NAFTA has been blamed for the rapid decline of American manufacturing and job losses as companies moved factories to lower-wage Mexico.

It’s also blamed for suppressing U.S. wages.

Mr. Trump’s tough talk on trade and NAFTA in particular helped fuel his support across the Rust Belt, which sealed his victory in November.

Tackling NAFTA followed a series of get-tough moves on trade by the Trump administration, including slapping tariffs on Canadian soft lumber exports and launching investigations of imported steel and aluminum.

The administration was ready to begin the exit from NAFTA within days. Under Article 2205, which allows any of the parties to unilaterally drop out, the exit would have been completed in six months.

Mr. Pena Nieto and Mr. Trudeau phoned the White House Wednesday evening to ask Mr. Trump to enter talks on NAFTA rather than quit, a move that would have left America’s neighbors in an economic lurch.

Mr. Trump said his “very good” relationships with Mr. Pena Nieto and Mr. Trudeau, as well as his respect for their countries, prompted him to give renegotiation a chance.

He said he told them “let’s see if we can make it a fair deal, because NAFTA has been a horrible deal for the United States. It’s been very good for Canada. It’s been very good for Mexico. But it has been horrible for the United States.”

Mexico and Canada gave a more subdued description of the call, but all the parties said the result was consensus on the need to “modernize” NAFTA.

“During the call, both Presidents spoke of the shared goal of modernizing the North American Free Trade Agreement,” said a statement form Mr. Pena Nieto’s office. “The leaders agreed on the advisability of maintaining the North American Free Trade Agreement and working with Canada to carry out a successful renegotiation for the benefit of the three countries.”

NAFTA hasn’t been all bad for the U.S. About one-third of all U.S. merchandise exports go to Mexico and Canada, and exports from American service industries and from the agriculture sector increased dramatically under the agreement, noted the conservative Club for Growth, which advocates free trade policies.

The proposed U.S. pullout also faced resistance from free trade Republicans in Congress.

“We should improve NAFTA, not abandon it,” Sen. Jeff Flake, an Arizona Republican and frequent critic of Mr. Trump, said on the Senate floor after Mr. Trump’s announcement.

He pointed to Arizona’s flourishing trade with Mexico, which exceeded $15 billion in 2016 and supports about 100,000 jobs in the Grand Canyon State.

“The bottom line is that trade is good for American businesses, it’s good for American workers, and it’s good for American consumers,” he said. “Trade is not a zero-sum game where one party wins and the other party loses. Free trade benefits everyone. I hope that we remember this as we look toward NAFTA’s future.”

Jason Grumet, president of the Bipartisan Policy Center, said Mr. Trump was taking the right approach in moving cautiously to revamp “a complex set of institutional and economic relationships that have developed over the past 25 years.”

“NAFTA has created millions of U.S. jobs that are now dependent on that trade. However, as technological advances and shifting global dynamics reshaped the business landscape, NAFTA has remained largely unchanged,” he said.

“We commend the administration’s intent to update and reform NAFTA to ensure that U.S. interests and workers are protected, and to bolster our competitiveness. However, we should remain cautious about taking extreme steps that could damage American businesses and the overall U.S. economy,” said Mr. Grumet.

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