Japan's Nikkei 225 index rose less than 0.1% to 23,811.54 and in Hong Kong the Hang Seng gained 0.2% to 27,949.64. Australia's S&P ASX/200 picked up 0.2% to 7,100.30 and the Sensex in India also rose 0.2%, to 41,473.97.
Markets were closed in Shanghai and the rest of mainland China, South Korea, Malaysia and Taiwan.
As authorities confirmed more cases of the new virus first reported in the central Chinese city of Wuhan, investors continued to monitor developments in the international effort to keep it from spreading further and potentially harming the global economy.
The World Health Organization decided Thursday against declaring the outbreak a global emergency for now. Such a declaration could increase resources for battling the outbreak but also result in trade and travel restrictions and other economic damage.
Fears that the coronavirus could spread have weighed on global markets this week, driving up demand for U.S. government bonds and safe-play stocks.
Market "traders are weighing the anticipated China growth fallout against the backdrop of the current global growth recovery. While the calculus is not coming up roses, it's far from a state of global market panic," Stephen Innes of AxiCorp said in a commentary.
"Still, if risk aversion starts to spread beyond China's borders and starts to affect more than the usual suspect's luxury, travel, and tourism, then we will likely see a more significant dive in the broader global indices," he said.
Major U.S. stock indexes closed mostly higher Thursday, as gains in technology and industrial companies offset declines elsewhere in the market.
The S&P 500 notched a small gain for the second straight day, climbing 0.1% to 3,325.54, while a modest pickup nudged the Nasdaq composite to an all-time high of 9,402.48, up 0.2%. The Dow Jones Industrial Average edged 0.1% lower to 29,160.09, its third straight day of losses as the benchmark was weighed down by a steep drop in shares of Travelers Cos.
The Russell 2000 index of smaller company stocks rose less than 0.1%, to 1,685.01.
Traders also had their eye on a mixed batch of company earnings reports, including encouraging quarterly results from American Airlines and Citrix Systems, and disappointing report cards from Travelers and Raymond James Financial.
"Today was driven a bit by earnings, but also by the coronavirus fears," said J.J. Kinahan, chief strategist with TD Ameritrade. "Asian markets had a really tough night and that was our lead-in, that put a bit of extra pressure on the market coming in."
Excluding the Nasdaq, the major U.S. stock indexes are on track to end the week with a loss.
Bond prices rose, pulling the yield on the 10-year Treasury lower to 1.73% from 1.77% late Wednesday.
Benchmark crude oil gained 14 cents to $55.73 per barrel in electronic trading on the New York Mercantile Exchange. It fell $1.15 to settle at $55.59 a barrel on Thursday. Brent crude oil, the international standard, picked up 18 cents to $62.22 per barrel. It dropped $1.17 to close at $62.04 a barrel overnight.
Gold fell back, losing $4.30 to $1,561.10. Silver shed 3 cents to $17.80 per ounce and copper fell 4 cents to $2.73 per pound.
The dollar rose to 109.52 Japanese yen from 109.49 yen on Thursday. The euro weakened to $1.1053 from $1.1056.