“We’re not backing down,” the president said at the White House on Monday, as he reeled off a familiar litany about trade deals that he said had driven out factories and deprived American workers of jobs.

But Mr. Trump did open the door to a compromise, at least with Canada and Mexico, which are in negotiations with the United States to revise the North American Free Trade Agreement. If the two countries agree to a “new & fair” Nafta, they could be exempt from the tariffs, Mr. Trump said in a tweet on Monday morning.

The debate over tariffs has become a litmus test for Mr. Trump, putting his longstanding suspicion of free trade up against the equally fervent support for it among Republicans and members of his own administration.

Inside the White House, the debate has pitted hard-liners like Peter Navarro, the president’s trade adviser, and Commerce Secretary Wilbur Ross against more pro-trade voices, like Mr. Trump’s chief economic adviser, Gary D. Cohn, who argue that the measure could disrupt international alliances and global supply chains.

Republican lawmakers have criticized tariffs as undercutting the $1.5 trillion tax cut that they worked in lock step to pass last year, saying tariffs are essentially a tax increase that would slow economic growth. On Monday, they floated the idea of congressional action to try to block tariffs, should the president impose them.

Representative Kevin Brady, Republican of Texas and chairman of the House Ways and Means Committee, circulated a letter Monday expressing concern over the tariffs. Senator Dan Sullivan, Republican of Alaska, said he spent the weekend speaking with members of Congress and “senior administration officials” about his opposition to the president’s plan.

“As you know the administration is split itself,” Mr. Sullivan said from the annual CERAWeek energy conference in Houston, noting that details of the tariffs remain sparse.

Mr. Trump has heard all sides’ arguments, but his view has remained steadfast, one White House official said.

Still, the official said, the president is mindful enough of the arguments against potentially tanking the stock market that he has been somewhat open to a move to narrow the scope and effects of the tariffs while avoiding the perception that he was relenting. That would echo the approach the administration took to winding down the president’s promises on the Deferred Action for Childhood Arrivals, or DACA, program, which has protected young immigrants brought illegally to the United States as children.

The unsettled nature of a final policy was magnified by a conversation on Sunday between Mr. Trump and Prime Minister Theresa May of Britain.

Ms. May, a person briefed on the call said, warned Mr. Trump how dangerous the tariffs would be. Mr. Trump disagreed, but concluded the conversation by telling Ms. May that he had not made a final decision on what to do.

The president originally announced that he wanted to put new tariffs into effect this week, but a legal review has not been concluded. On Sunday, Mr. Navarro said the tariff announcement could come this week or the following week at the latest. He also reaffirmed that companies might be able to seek exemptions for certain foreign products they need for their business, a process likely to lead to months of furious lobbying.

A coalition of 25 industry associations representing farmers and companies that use steel and aluminum has started lobbying the administration and lawmakers, arguing that the tariffs are far broader than necessary and would create higher prices on American companies that buy and use metals.

The president and his advisers have repeatedly maintained that any tariffs will be imposed on imports from all countries without exception. The Commerce Department has concluded that imports of steel and aluminum pose a threat to national security, a determination that gives the president broad authority under United States law to impose tariffs.

But that legal standing will undoubtedly be challenged by other countries and companies, both in court and at the World Trade Organization, which requires that members treat all other members equally on trade. Creating exceptions for countries like Canada and Mexico for non-national security reasons like Nafta could invite challenges at the World Trade Organization, said Jennifer Hillman, a professor at Georgetown University Law Center.

“Unequivocally, I think there will be cases filed at the W.T.O., and there is plenty of ground to challenge this,” Ms. Hillman said.

On Monday, Robert Lighthizer, the United States trade representative, said Mr. Trump decided to link a tariff exemption to a revised Nafta deal “a couple of days ago” as Mr. Lighthizer prepared to travel to Mexico City to meet with his Mexican and Canadian counterparts.

“It makes sense, since this is a major irritant, to have it be considered,” Mr. Lighthizer said in Mexico City after meeting with Economy Minister Ildefonso Guajardo of Mexico and Foreign Minister Chrystia Freeland of Canada. He said the situation could be modified for Mexico and Canada given a successful Nafta renegotiation, as well as “perhaps other countries in other contexts where we have those kinds of problems.”

Even if Canada and Mexico were exempted from the tariffs of 25 percent on steel and 10 percent on aluminum, officials said the United States would impose a quota on those countries’ exports to prevent them from being used as a conduit for metals shipments from other nations.

Neither Canada nor Mexico appeared mollified by the prospect of a tariff exemption in exchange for bending to United States demands on Nafta. Ms. Freeland reiterated comments that any action that ensnared Canada would be “completely unacceptable.”

“México shouldn’t be included in steel & aluminum tariffs,” Mr. Guajardo said in a tweet. “It’s the wrong way to incentivize the creation of a new & modern #NAFTA.”

Michael Camuñez, a former Commerce Department official who advises international firms doing business in Mexico, called the proposed tariffs and the tweet a “terrible development” for the Nafta negotiations “because all parties have indicated that they will not negotiate with a gun against their head.”

The talks in Mexico City produced no meaningful progress, and ultimately would be decided by one man — Mr. Trump, Mr. Camuñez said. Negotiators have continued to clash over provisions in the pact, including rules for auto manufacturing, and the United States has continued to insist on changes that its trading partners say are nonstarters.

Mr. Camuñez said Mexico and Canada would retaliate “with good reason” if the steel and aluminum tariffs were imposed, an outcome that could lead to a breakdown in negotiations. “That would escalate tensions and that could lead the president to conclude that these negotiations aren’t going anywhere and blame the Mexicans and the Canadians,” Mr. Camuñez said.

Mr. Trump has threatened additional retaliation if other nations erect their own trade barriers, saying in a tweet over the weekend that the United States would make it harder for the European Union to sell cars in the country if it imposes tariffs on American imports.