The prospect for a possible easing of tensions between the two largest economies buoyed shares in China, where the Shanghai Composite surged 2.6% to 2,925.72 while the smaller Shenzhen A-share index leaped 3.7%.
In Europe, France's CAC 40 advanced 0.8% to 5,423, while Germany's DAX rose 1.2% to 12,191. Britain's FTSE 100 was up 0.4% at 7,404. U.S. shares were set to drift higher with Dow futures and the broader S&P 500 futures up 0.4%.
In an interview with CNBC, Trump said he expects to meet with Chinese President Xi Jinping at the Group of 20 summit in Japan later this month. He also said he was prepared to push ahead with plans to expand tariffs to encompass another $300 billion of Chinese exports to the U.S.
The positive momentum from Monday's rally carried over into Asia and beyond. Japan's benchmark Nikkei 225 rose 0.3% to finish at 21,204.28. Australia's S&P/ASX 200 added 1.6% to 6,546.30. South Korea's Kospi gained 0.6% to 2,111.81. Hong Kong's Hang Seng stood at 27,789.34, up 0.8%.
On Wall Street, shares continued their winning streak for a fifth day on Monday. That follows the strongest week for stocks since November in a marked turnaround for the market after escalating trade tensions caused prices to skid in May.
Jingyi Pan, market strategist at IG in Singapore, warned against excess optimism about ending the longstanding and increasingly bitter standoff between Beijing and Washington. "The seemingly complicated matter of a U.S.-China trade deal appears highly unlikely to be resolved from a high-level meeting between the two presidents," Pan said.
"Given the return of the U.S.-China tariffs threat to the table, look to Asia markets to waffle along." ENERGY: Benchmark U.S. crude rose 38 cents to $53.64 a barrel in electronic trading on the New York Mercantile Exchange. It slid 1.4% to $53.26 a barrel on Monday. Brent crude oil, the international standard, was up 4 cents at $62.33 a barrel.
CURRENCIES: The dollar rose to 108.70 yen from 108.44 yen on Monday. The euro was flat at $1.1310.