2nd March 2018, 21:30
EU trade chiefs are considering slapping 25% tariffs on around $3.5bn (£2.5bn) of imports from the US, Reuters news agency reports.
World Trade Organization Director General Roberto Azevedo said: "A trade war is in no one's interests."
The rhetoric ramped up as Mr Trump tweeted that "trade wars are good".
International condemnation has greeted the US president's Thursday announcement that he plans to impose a 25% tariff on steel imports and 10% on aluminum next week.
What are EU officials saying?
The European Union is reported to be considering retaliatory tariffs, targeting US steel, agriculture and other products.
European Commission head Jean-Claude Juncker promised to react firmly.
"We will not sit idly while our industry is hit with unfair measures that put thousands of European jobs at risk," he said.
Speaking to a German TV programme, he vowed: "We will put tariffs on Harley-Davidson, on bourbon and on blue jeans – Levi's."
French economy minister Bruno Le Maire said there would "only be losers" in a US-EU trade war.
Mr Le Maire vowed a "strong, co-ordinated and united response from the EU".
Canada, Mexico, China and Brazil have also said they are considering retaliatory steps.
What does Trump administration say?
Mr Trump tweeted on Friday morning that the US was "losing billions of dollars on trade" and would find a trade war "easy to win".
When a country (USA) is losing many billions of dollars on trade with virtually every country it does business with, trade wars are good, and easy to win. Example, when we are down $100 billion with a certain country and they get cute, don’t trade anymore-we win big. It’s easy!
Critics argue that the tariffs would fail to protect American jobs and ultimately raise prices for consumers.
But US Commerce Secretary Wilbur Ross used a can of Campbell's Soup to defend the tariffs on Friday morning as "no big deal".
"In a can of Campbell's Soup, there are about 2.6 pennies worth of steel," he told CNBC.
"So if that goes up by 25%, that's about six-tenths of one cent on the price on a can of Campbell's Soup.
"I just bought this can today at a 7-Eleven and it priced at a $1.99. Who in the world is going to be too bothered?"
The US buys 15% of UK steel exports
Are trade wars good?
By Theo Leggett, business correspondent
If trade wars really were good and easy to win, the World Trade Organisation probably wouldn't exist.
Most countries believe that negotiations are best carried out and disputes settled through a rules-based system. Introducing trade barriers on a tit-for-tat basis has the potential to harm companies on both sides.
But that's unlikely to bother Mr Trump. His campaign rhetoric drew heavily on the perceived threat to traditional US industries from foreign interlopers acting unfairly. He's simply continuing in that vein.
And it's unlikely to register much with the steelworkers of Pennsylvania and Indiana. Concerned about their jobs and the future, many will welcome Mr Trump's comments.
What are the stakes for US?
Mr Trump has lamented the decline of the US steel industry, which since 2000 has seen production drop from 112m tons to 86.5m tons in 2016.
The number of employees working in the sector has fallen over the same period from 135,000 to 83,600.
But experts say far more Americans work in industries that depend on steel products, than are employed in steel plants.
Steel mills in 2015 employed about 140,000 Americans, according to census data.
But 6.5 million Americans work for manufacturers who make things using steel.