Other countries may also avoid import taxes amid reported concern in administration about national security alliances.

Donald Trump could exempt Canada, Mexico and other countries from stiff import taxes he is expected to impose this week, the White House said on Wednesday.

The US president and his allies previously indicated that there would be no exceptions to the trade tariffs, which have been condemned by the European Union (EU) and led to the resignation of the president’s chief economic adviser, Gary Cohn.

“We expect the president will sign something by the end of the week and there are potential carve-outs for Mexico and Canada based on national security, and possibly other countries as well, based on that process,” the White House press secretary, Sarah Sanders, told reporters on Wednesday.

“That would be a case-by-case and country-by-country basis but it would be determined whether or not there is a national security exemption.”

The secretary of state, Rex Tillerson, and the defense secretary, Jim Mattis, have expressed concern that the tariffs could undermine vital national security alliances, according to media reports. But the White House has taken a hard line until now.

Peter Navarro, the hardline author of Death By China and director of the White House National Trade Council, told CNN’s State of the Union on Sunday: “As soon as you start exempting countries, you have to raise the tariffs on everybody else. As soon as you exempt one country, then you have to exempt another country. And so it’s a slippery slope.”

Pressed on Canada’s claim that it should be exempted, Navarro added: “If you exempt Canada, then you have to put big, big tariffs on everybody else.”

But on Tuesday, Trump raised the prospect of making exceptions for Canada and Mexico if the North American Free Trade Agreement (Nafta) can be successfully renegotiated. The EU, meanwhile, threatened to complain to the World Trade Organisation and slap retaliatory tariffs on American imports including bed linen, chewing tobacco, cranberries and orange juice.

Trump’s overall commitment to the tariffs remains firm. He tweeted early on Wednesday: “From Bush 1 to present, our Country has lost more than 55,000 factories, 6,000,000 manufacturing jobs and accumulated Trade Deficits of more than 12 Trillion Dollars,” the president tweeted. “Last year we had a Trade Deficit of almost 800 Billion Dollars. Bad Policies & Leadership. Must WIN again! #MAGA”.

Cohn, a former Goldman Sachs president and free market advocate, became the latest White House figure to quit on Tuesday. Sanders declined to be drawn on his possible successor. “I’m not going to get into any naming or a list but I can tell you the president has a number of people under consideration and he’s going to take his time making that decision.”

Cohn was regarded as a “globalist” by economic nationalists, who now appear to be in the ascendent as they appeal to Trump’s protectionist instincts. Senior Republicans, for whom free trade is an article of faith, have expressed concern over who now has the president’s ear.

But Sanders insisted at the daily briefing: “The president’s got a number of very accomplished, smart, capable people around him and he is going to continue to lean on a lot of those people but at the end of the day, the American people voted overwhelmingly for President Donald J Trump.

“They voted for his policies, his agenda and for him to be the ultimate decision-maker and I think that everyone can rest assured in the American people’s choice on that front and they’ve made the right one.”

The press secretary also denied that Cohn’s departure, which came on the heels of the communications director Hope Hicks’s resignation last week, demonstrated chaos in the west wing. “Look, this administration has had a historic first year. We’re going to continue to do great things. This is an intense place, as is every White House, and it’s not abnormal that you would have people come and go, but we’re continuing to do great work, we’re continuing to focus on the president’s agenda and that’s what we’re all here to do.”

She added: “There are historic things that have taken place in the first year. Sounds like a very functioning place of business to me.”

Other officials also denied that Cohn’s departure left the president’s economic team in disarray. Steven Mnuchin, the treasury secretary and another Goldman Sachs veteran who is part of the diminished “globalist” wing of the administration, told Fox Business Network: “I’ve known Gary for a long time. He has been a terrific partner in working here on tax reform and other things.

“As you know, there was a lot of speculation after tax reform whether he’d stay or not and Gary decided to move on. But we have a terrific economic team here. And we’re going to consistently execute the president’s strategy.”

Mnuchin defended Trump’s planned tariffs – 25% on steel and 10% on aluminium – but raised the prospect of exceptions for certain countries. “The president is determined that we maintain the steel industry. This is not new. Other presidents have done similar types of things. So let me just remind you, this is not the first time this has ever been done.

“And we’ve done a careful analysis and looking at this. And as I said, we have a mechanism to carve out countries. We’ll consider these things. And we’re moving forward with the ability to do this on steel and aluminum and move forward with our strategy.”

Democrats have been less critical of Trump’s tariff proposal, faulting the execution but not the instinct.

The Democratic Senate leader Chuck Schumer told reporters on Wednesday that he agrees “wholeheartedly” with Trump’s desire to go after China but faulted the White House for a plan that he said hurts trading partners who are not “rapacious in their trading” such as Canada and countries in western Europe. Schumer urged Trump to follow through on his threat to go after China.

He said: “They ought to not let all of these academics who just say anytime you try to do anything on trade it’s protectionist. They ought to not let that deter them. But they ought to put together a real plan that works.”

Cohn’s departure represents another blow to the administration, which has suffered several high-profile resignations since the election. They include Trump’s former chief of staff, Reince Priebus, and his former chief strategist, Steve Bannon.