Republicans have raised concern about the US president's plan to impose tariffs on metals, with the party's top lawmaker calling for it to be scrapped.
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."
What does Trump want to do and why?
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.
We are on the losing side of almost all trade deals. Our friends and enemies have taken advantage of the U.S. for many years. Our Steel and Aluminum industries are dead. Sorry, it’s time for a change! MAKE AMERICA GREAT AGAIN!— Donald J. Trump (@realDonaldTrump) March 5, 2018
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.
The United States has an $800 Billion Dollar Yearly Trade Deficit because of our “very stupid” trade deals and policies. Our jobs and wealth are being given to other countries that have taken advantage of us for years. They laugh at what fools our leaders have been. No more!— Donald J. Trump (@realDonaldTrump) March 3, 2018
Has Trump got political support at home for a trade war?
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".
Responding to the criticism on CNN, White House economic adviser Peter Navarro dismissed the number of Republicans opposed Mr Trump's tariff plan.
"Guess what: He beat them," Mr Navarro said, referring to the outcome of the 2016 election.
He added that Republicans are "dead wrong on the economics".
Industry bodies like the US Motor and Equipment Manufacturers Association have expressed deep concern.
How has the EU responded to the tariff threat?
Levi jeans and bourbon could be hit with a 25% import tax by the European Union if Mr Trump imposes tariffs on European steel and aluminium.
Cecilia Malmström, EU Commissioner for Trade, told the BBC the items were on a draft list of US goods to be taxed.
Ms Malmström told the BBC: "We are looking at possibilities to retaliate, meaning we will also put taxes or tariffs on US imports to the European Union."
What do US trading partners make of this?
Canadian Prime Minister Justin Trudeau had "forcefully defended" Canadian industry in a phone call to Mr Trump on Monday, his office said.
A spokesman described the conversation as "constructive" but gave no details. Canada has warned that tariffs would cause disruption on both sides of the border.
Downing Street said that during Mrs May's call to President Trump on Sunday she raised "our deep concern at the president's forthcoming announcement on steel and aluminium tariffs, noting that multilateral action was the only way to resolve the problem of global overcapacity in all parties' interests".
Zhang Yesui, spokesperson for China's National People's Congress, said it was natural that "some friction will exist" between the US and China, given the volume of trade between them surpassed $580bn (£420bn) last year.
But he said China would take "necessary measures" if its interests were hurt.
EU trade chiefs could apply 25% tariffs on about $3.5bn of imports from the US - targeting iconic US exports including Levi's jeans, Harley-Davidson motorbikes and bourbon whiskey.
Brazil, Mexico and Japan have said they will consider retaliatory steps if the president presses ahead with his plan next week.
The move has also been strongly criticised by the International Monetary Fund and the WTO.
Is Trump right about the trade imbalance?
The US imports steel from more than 100 nations and brings in four times more steel from abroad than it exports.
Since 2000, the US steel industry has suffered, with production dropping and the number of employees in steel work falling.
The US is the largest export market for EU cars - making up 25% of the €192bn (£171bn; $237bn) worth of motor vehicles the bloc exported in 2016 (China was second with 16%).