Benchmarks in Tokyo, Hong Kong and Sydney fell more than 1% while Shanghai and South Korea also retreated.
Trump said he has "no deadline" for a trade deal and doesn't mind waiting until after next year's presidential election to make one. Investors had hoped for at least enough progress to stave off new U.S. tariffs on Chinese goods, including smartphones and laptops, scheduled to start Dec. 15.
Tensions flared last week after Trump signed legislation expressing U.S. support for pro-democracy demonstrators in Hong Kong.
"Until both sides dial down their hawkish rhetoric, markets will continue to pull back earlier optimism," said DBS Group analysts in a report. "Trade war will be the key driver of sentiment in the immediate few weeks."
Tokyo's Nikkei 225 index sank 1.03% to 23,141.68 and Hong Kong's Hang Seng gave up 1.2% to 26,077.77.
The Shanghai Composite Index retreated 0.3% to 2,874.88 and Seoul's Kospi declined 0.8% to 2,060.81. Australia's S&P-ASX 200 fell 1.4% to 6,621.40 and benchmarks in Taiwan, New Zealand and Southeast Asia also retreated.
India's Sensex bucked the trend, opening up 0.1% to 40,724.00.
On Wall Street, the benchmark S&P 500 index fell 0.7% to 3,093.20. The Dow Jones Industrial Average lost 1% to 27,502.81. The Nasdaq dropped 0.6% to 8,520.64.
Negotiators have yet to agree on details of an interim deal Trump has called "Phase 1."
Investors also are weighing the potential for additional trade disputes.
On Tuesday, Trump proposed tariffs on $2.4 billion in French products in retaliation for a tax on global tech giants including Google, Amazon and Facebook. That follows a threat Monday to raise tariffs on steel and aluminum from Argentina and Brazil.
Also Wednesday, Australia disappointed investors by reporting third-quarter economic growth declined to 0.4% over the previous quarter from 0.6% in the three months ending in June. The Australian central bank had described the economy as experiencing a "gentle upturn."
ENERGY: Benchmark U.S. crude rose 42 cents to $56.52 per barrel in electronic trading on the New York Mercantile Exchange. The contract gained 14 cents on Tuesday to close at $56.10. Brent crude, used to price international oils, rose 49 cents to $61.31 per barrel in London. It lost 10 cents the previous session to $60.82.
CURRENCY: The dollar declined to 108.58 yen from Tuesday's 108.63 yen. The euro edged down to $1.1077 from $1.1084.