An escalating war of words between the EU and the U.S. first began when Trump announced plans to slap hefty tariffs on steel and aluminum imports. He said "very stupid" trade deals had allowed countries from all over the world to take advantage of the world's largest economy.
That prompted a firm response from the U.S.'s trading partners, as well as criticism from the International Monetary Fund (IMF) and the World Trade Organization (WTO).
"I'm suggesting that we stop playing tit-for-tat, that we get our blood pressure to go back down to normal (and) that we sit down at the table and find a way to resolve this issue," Fiat Chrysler CEO Sergio Marchionne said Tuesday.
"I don't think we have to escalate this into a full-blown trade war," he added.
Shortly after the U.S. president's full-throated backing of tariffs, EU trade chiefs were thought to be considering whether or not to impose a 25 percent tax on around $3.5 billion of U.S. imports, Reuters reported.
European Commission President Jean-Claude Juncker had previously vowed to react firmly to Trump's tariff threats.
"If the E.U. wants to further increase their already massive tariffs and barriers on U.S. companies doing business there, we will simply apply a tax on their cars which freely pour into the U.S.," Trump said on Twitter.
That could spell trouble for European carmakers given that several of the continent's most popular brands have a major manufacturing footprint in the U.S.
"Free and fair trade is the best for all societies and at the end of the day it delivers win-win situations. I hope the politicians really think about it and create win-win situations for their societies," Ralf Speth, CEO at Jaguar Land Rover, told CNBC on Tuesday.
Speaking at the Geneva Motor Show in Switzerland, Speth said it was impossible to be concerned about the prospect of a brewing trade war at this stage because "hardly any" information on how it would play out was available.
EU and U.S. auto-related trade account for around 10 percent of total trade between the two regions, according to the latest data from the European Automobile Manufacturers' Association (ACEA).
The Brussels-based lobbying group also said that the U.S. is the number one destination for EU car exports both in terms of units (approximately a 20 percent share) and value (almost a 30 percent share).
"If you really think this through, I think everybody will realize that free trade and openness is beneficial for everybody," Hakan Samuelsson, CEO at Volvo Cars, told CNBC on Tuesday.
When asked to what extent it would likely damage transatlantic business operations for the Swedish carmaker, Samuelsson said: "It would be very bad, of course, but I think it is too early to speculate about that."
On Monday, the CEO of VW Group said he was "calmly assessing" the situation and would be prepared to take the "necessary decisions." Matthias Mueller added there was no need to react to the threat of additional taxes on European car sales to U.S. customers at this moment.