White House spokeswoman Sarah Sanders told a media briefing "there are potential carve-outs for Mexico and Canada based on national security, and possibly other countries as well".
Trump has already frozen years of talks on a trans-Atlantic market opening agreement, pulled out of a trans-Pacific trade accord and demanded changes to a 24-year-old commercial pact among the U.S., Canada and Mexico.
On March 1, Trump said the United States will impose tariffs of 25 percent on steel and 10 percent on aluminum this week, a step he says is meant to revive domestic manufacturing, but the announcement has sparked worries about a global trade war.
In response to the planned steel and aluminum tariffs, the European Union has proposed tariffs on US exports including motorcycles and bourbon.
"Gary brought a strong growth orientation for the economy and a very balanced perspective, particularly on critical topics like trade, " said Rich Lesser, chief executive of the Boston Consulting Group, who served on one of Trump's business advisory panels past year and attended policy meetings with Cohn.
They said the administration should also regularly review the measure to determine whether it is having the intended effect.
Prime Minister Justin Trudeau said he wants to withhold judgment until the final details are out. "It's not precise. And as a result, it could cause a mess of collateral damage that hurts America more than it helps", Schumer said on Wednesday...
A trade official quoted Canada's WTO ambassador as saying: "We fear that the United States may be opening a Pandora's Box that we would not be able to close".
Equities also got a bid earlier after White House trade adviser Peter Navarro, who favors strong tariffs, said in a Bloomberg interview he was not a candidate to replace Cohn.
White House officials said the tariffs dispute contributed to Cohn's decision to leave but that it was not the sole reason.
More than 100 members of President Trump's own party signed a letter addressed to him Wednesday, expressing "deep concern" over his proposed tariffs and asking him to reconsider.
The dollar's recovery was stoked by Wednesday's data on US private hiring and labor costs that reinforced the view of underlying strength in the economy.
"What good has ever come from trade wars that result in a decrease in economic activity?"
"A trade war has no winners and if it does not happen, for the better, then we can work with our American friends and other allies on the core issue of this problem, overcapacity", European Commissioner for Trade Cecilia Malmstrom said.
The chief of the International Monetary Fund, Christine Lagarde, on Wednesday in a European radio interview, warned of a global trade war, predicting the US tariffs could lead to "a drop in growth, a drop in trade, and it will be fearsome".
Mr Cohn wouldn't agree to publicly back the president's tariffs, the people said.
Trump is pushing the tariffs as a national security necessity, but also as a way to revitalize the US steel industry, which employs around 140,000 people.
"In a so-called trade war ... nobody wins, one generally finds losers on both sides", International Monetary Fund head Christine Lagarde said on Wednesday, adding that a trade war would take a "formidable" toll on global economic growth.