US Speaker of the House Paul Ryan said he was “extremely worried” about the impact of a trade war, adding that it could undermine economic gains.
But Mr Trump pushed back during a White House meeting with Israeli Prime Minister Benjamin Netanyahu.
“We’re not backing down,” he told reporters on Monday.
“I don’t think you’re going to have a trade war,” he said.
White House press secretary Sarah Huckabee Sanders said on Monday that President Trump was “very confident” the US would win any trade war.
Mr Trump’s Monday comments came an hour after Mr Ryan released a statement urging the White House to reconsider its plan.
“We are extremely worried about the consequences of a trade war and are urging the White House to not advance with this plan,” Mr Ryan’s spokeswoman AshLee Strong said.
“The new tax reform law has boosted the economy and we certainly don’t want to jeopardise those gains.”
Mr Trump’s announcement last week that he would tax imported steel and aluminium has prompted worldwide reaction.
The World Trade Organization (WTO) on Monday also called on member states to “stop the fall of the first dominoes” of a trade war.
“Once we start down this path it will be very difficult to reverse direction,” WTO Director General Roberto Azevedo told negotiators in Geneva on Monday. “An eye for an eye will leave us all blind and the world in a deep recession.”
Earlier on Monday, Mr Trump hinted that if the US achieved a better deal for itself in the North American Free Trade Agreement (Nafta) he would abandon plans for tariffs on US neighbours.
Those tariffs could be removed for Canada and Mexico if they signed a “new and fair” agreement, he suggested.
The current round of Nafta talks, which focus on updating the 24-year old treaty, are due to finish on Monday and have achieved little.
Mr Trump has decried the US trade deficit with other countries, which he has blamed on “‘very stupid’ trade deals and policies”.
On Thursday, he said steel imports would face a 25% tariff and aluminium 10%.
He issued a threat against EU-made cars on Saturday, which he repeated during his Oval Office meeting with the Israeli prime minister on Monday.
“They have trade barriers that are worse than tariffs. They also have tariffs by the way, but they have trade barriers far worse than tariffs.”
“And if they want to do something we’ll just tax their cars that they send in here like water,” he vowed.
Over the weekend some Republicans began to question the tariff proposal and have urged the president to reconsider.
Many argue that the impact of tax cuts that were passed earlier this year will be wiped out as countries levy new tariffs on US goods and the price of metals climbs.
Senator Orrin Hatch said American citizens would be made to pay, adding that Mr Trump’s “action could very well undercut the benefits of the pro-growth tax reform we fought to get on the books”.
Senator Ben Sasse agreed that “kooky 18th Century protectionism will jack up prices on American families”. (Courtesy BBC)