Traders were awaiting talks between central bankers and other financial leaders of the Group of Seven industrial nations on how to tackle the slowdown brought on by the outbreak that began in China and has spread to dozens of countries, killing about 3,100 people and sickening more than 90,000.
Japan's Nikkei 225 lost 1.2% to finish at 21,082.73 after gaining in the morning. Australia's S&P/ASX 200 rose 0.7% to 6,435.70 after the Reserve Bank of Australia cut its key interest rate to a record-low 0.5%.
South Korea's Kospi rose 0.8% to 2,019.11. Hong Kong's Hang Seng was little changed but inched up less than 0.1% to 26,298.51, while the Shanghai Composite advanced 0.6% to 2,988.81.
India's Sensex was little changed after rising earlier in the day, while Taiwan's benchmark surged 1.4%.
But the mood shifted in Tokyo by midday, as thoughts turned to what the Bank of Japan might be able to do to help counter the slowdown worsened by the outbreak of the new virus that causes a disease called COVID-19. The BOJ's policy rate has stood at minus 0.1% for several years and the central bank has been purchasing tens of billions of yen (billions of dollars') worth of government bonds and other assets to help keep credit cheap and stave off deflation as the population in the world's No. 3 economy ages and shrinks.
Shares of manufacturers fell as the Japanese yen gained against the U.S. dollar, potentially hurting exports. The dollar was trading at 107.79 Japanese yen, down from 108.27 yen on Monday. The euro strengthened to $1.1149 from $1.1133.
But elsewhere in the region the mood was upbeat after the Dow Jones Industrial Average soared nearly 1,300 points, or 5.1%, to 26,703.32 on Monday, its biggest-ever point gain and the biggest percentage increase since March 2009. The huge gains clawed back some of the ground lost last week in a massive sell-off that gave stocks their worst stretch since the financial crisis of 2008.
"So why are markets so pumped by prospects of monetary response; arguably not the most apt tool to address the direct fallout from coronavirus related disruptions?" said Vishnu Varathan at Mizuho Bank in Singapore.
"One reason may be that more nuanced measures to ease cash-flow will offer a reprieve for businesses and households affected by seizures in activity and disruptions in supply chains," he said.
Technology companies led the broad gains in New York, where the S&P 500 index jumped 4.6% to 3,090.23 in its best day since December 2018. Apple climbed 9.3% and Gilead Sciences rose 8.7%. The biotechnology company has been testing one of its drugs as a potential treatment for the coronavirus.
The Nasdaq added 4.5% to 8,952.16. The Russell 2000 index of smaller company stocks picked up 2.9% to 1,518.49.
Even with Monday's big rally, the major U.S. indexes remain in the red for the year.
The virus epidemic that began in central China has been shutting down industrial centers, emptying shops and severely crimping travel all over the world. More companies are warning investors that their finances will take a hit because of disruptions to supply chains and sales.
Amid the worsening outlook, investors are increasingly anticipating that the Federal Reserve and other major central banks around the world will lower interest rates or take other steps to shield the global economy from the effects of the outbreak.
"Investors have convinced themselves that global central banks will likely be even more accommodative in order to short-circuit any psychological damage, " said Sam Stovall, chief investment strategist at CFRA.
Bill Nelson, chief economist at the Bank Policy Institute and a former Fed economist, said the Fed and other major central banks, possibly including China's, could announce coordinated rate cuts by Wednesday morning. The cut would at least be a half-point and perhaps even three-quarters, he said.
"The only way to get a positive market reaction is to deliver more than expected," he said.
The International Monetary Fund and World Bank announced simultaneously Monday that they are ready to help countries affected by the coronavirus through their emergency lending programs and other tools.
And Christine Lagarde, the head of the European Central Bank, said Monday that Europe's top monetary authority is ready to take "appropriate and targeted measures" if necessary to support the economy against the headwinds from the new coronavirus.
U.S. Treasury Secretary Steven Mnuchin and Federal Reserve Chairman Jerome Powell will lead the call Tuesday. The group includes Japan, Germany, Britain and France, among others. The G-7 often issues statements pledging cooperation amid global economic turbulence.
The Organization for Economic Development, a research organization made up of mostly advanced economies, said Monday that the viral outbreak "presents the global economy with its greatest danger since the financial crisis" in 2008.
The OECD cut its world growth forecast and said that even if there are only limited outbreaks outside China, the global economy will grow just 2.4% this year, the weakest since the crisis. That forecast matches several private estimates.
If other countries are hit with outbreaks similar to China's, growth could fall as low as 1.5%, the OECD said.
Benchmark U.S. crude rose 68 cents to $47.43 a barrel. It jumped $1.99, or 4.4%, to $46.75 per barrel on Monday. Brent crude, the international standard, gained 53 cents to $52.43.