The moves followed a selloff on Wall Street after California Governor Gavin Newsom ordered bars closed and restaurants and movie theatres to cease indoor operations.
S&P 500 futures were 0.3% stronger after the index lost 0.9% on Monday.
Tension grew between Washington and Beijing after the United States rejected China's claims to offshore resources in most of the South China Sea.
The Trump Administration also plans to scrap a 2013 auditing agreement that could foreshadow a broader crackdown on U.S.-listed Chinese firms.
"It's not just the tempo which is picking up, but the aspect of so many areas being pulled into the dispute," said Vishnu Varathan, head of economics at Mizuho Bank in Singapore.
"Last time, it was really about the bottom line," Varathan said, but what had primarily been a trade dispute is now much broader, making a resolution less likely and the next moves less predictable.
The Shanghai index fell 0.7% despite official figures showing Chinese exports and imports topped forecasts in June, while China continued to buy significant amounts of commodities, including iron ore.
The return of restrictions in California also has markets on edge about whether the coronavirus can wreak more economic harm. Global infections have surged by a million in five days, and top 13 million.
ALL ABOUT 2021
Oil prices, a proxy for global energy consumption and growth expectations, reflected the concerns. U.S crude futures fell 1% to $39.70 per barrel and Brent futures fell 0.8% to $42.40 per barrel.
Investors sought out the safety of euro zone government bonds. Most yields were two to three basis points lower, with Germany's 10-year Bund yield, the region's benchmark, dropping to -0.42%.
There are also signs of an interruption to the steady flow of better-than-expected economic data. On Tuesday, data showed Singapore entered recession last month, with the economy contracting 41.2% for the quarter, worse than the 37.4% analysts had forecast.
Britain's gross domestic product rose by 1.8% in May after falling by a record 20.8% in April, the Office for National Statistics said, well below forecasts in a Reuters poll.
Currency markets kept the dollar in a tight range. Against a basket of currencies, the dollar index was down 0.1% at 96.453 =USD . The euro was 0.2% stronger against the dollar at $1.1369.
The focus will shift later to U.S. earnings, with JP Morgan, Citigroup, Wells Fargo and Delta Air Lines due to report on Tuesday to a market already looking to 2021 and beyond.
"It's really about 2021 -- 2020 is over," said fund manager Hugh Dive, chief investment officer at Atlas Funds Management in Sydney, where earnings season begins next month.
"The outlook statements are what the market will look at," he said. "If a company surprises on the upside with their 2020 earnings, but has shaky commentary for 2021 ... they're not going to be rewarded for that."
Spot gold slipped 0.5% at $1,793.14 per ounce.
(Additional reporting by Pete Schroeder in Washington and Paulina Duran in Syndey; editing by Larry King) ((Tom.Arnold@thomsonreuters.com; +442075428510; Reuters Messaging: firstname.lastname@example.org))