Fears the spat between the world's two largest economies could intensify added to pressure on oil prices, which extended Friday's big fall into the start of week, while the dollar retreated from a seven-month high against a basket of currencies, Reuters reported.
The MSCI world equity index, which tracks shares in 47 countries, fell 0.3 percent, nearing a seven-day low.
US stocks fell at the open on Monday. The Dow Jones Industrial Average fell 208.44 points, or 0.83 percent, at the open to 24,882.04. The S&P 500 opened lower by 18.55 points, or 0.68 percent, at 2,761.11. The Nasdaq Composite dropped 53.82 points, or 0.69 percent, to 7,692.55 at the opening bell.
Trump announced tariffs on Friday on $50 billion of Chinese imports, including cars, starting on July 6.
China said it would retaliate immediately by slapping duties on American export products, including crude oil, and suspend all previous trade agreements with Trump's administration.
The exchange of blows between Washington and Beijing has heightened fears of a protracted dispute that could hurt global growth and particularly Europe, given that Trump has signaled he wants to impose tariffs on automotive exports.
Those concerns saw European bourses start the week in the red, after Asian shares fell early on Monday to a 2-1/2 week low.
Futures on main euro zone benchmarks were trading down 0.2-0.5 percent as investor angst about the outlook for economic growth filtered through to European stocks.
The pan-regional STOXX 600 was on track to relinquish gains recorded on Thursday when a dovish European Central Bank pushed back expectations for an interest rate hike. (.STOXX)
Germany's DAX (.GDAXI) was down 1.36 percent while France's CAC 40 (.FCHI) declined 1.23 percent.
"This all shows how quickly trade tensions could escalate between the US and China," said Derek Halpenny, European head of global markets research at MUFG Bank.
"It may not be the end of the matter as US officials are looking at another $100 billion of Chinese imports on which they could impose tariffs if desired," he said.
A potentially destabilizing vote in German Chancellor Angela Merkel's governing coalition partner over a migration plan weighed on the euro and could put further pressure on European shares.
In commodity markets, Brent crude futures (LCOc1) fell to a six-week low of $72.45 a barrel on Monday after reports that top suppliers Saudi Arabia and Russia would likely agree to increase production at the June 22 OPEC meeting in Vienna.
US light crude oil (CLc1) hit a two-month low of $63.59 a barrel before recovering somewhat to trade at $64.72, down 30 cents, by 1100 GMT.
China has hiked its list of US goods on which it said it would slap tariffs six-fold from a version released in April, but the value was kept at $50 billion, as some high-value items such as commercial aircraft were deleted.
The immediate fallout from the dispute was limited in currency markets although the escalation appeared to encourage some risk aversion as the safe haven Japanese yen recovered from three week lows against the dollar. (JPY=D3)