Decades of trade integration across North America are on the precipice of major disruption by tariffs that President Trump said he wants to impose on Canada and Mexico, the United States’ top trading partners, as soon as Saturday.

And while tariffs are predicted to inflict pain on all three nations, they would cause more damage to Canada and Mexico, smaller economies that are deeply dependent on the United States.

Officials in both countries breathed a brief sigh of relief when Mr. Trump stopped short of making tariffs part of his blizzard of executive orders early on his first day in office. But the relief was short-lived: later that evening, Mr. Trump told reporters he was still planning to pursue tariffs.

“We’re thinking in terms of 25 percent on Mexico and Canada,” Mr. Trump said in the Oval Office. “I think we’ll do it February 1.”

In his confirmation hearing on Wednesday, Mr. Trump’s nominee to run the Commerce Department, Howard Lutnick, said Canada and Mexico could avoid a first wave of tariffs if they acted to address Mr. Trump’s demands to better manage the flow of undocumented migrants to the United States and curb the export of fentanyl.

“As far as I know, they are acting swiftly, and if they execute it, there will be no tariff,” he said. “And if they don’t, then there will be.”