Japan's benchmark Nikkei average, however, dropped 0.61% as telecom stocks fell after Nippon Telegraph and Telephone Corp announced a $38 billion take-private of its wireless carrier business, paving the way for price cuts in the sector. Shares going ex-dividend was also expected to dampen market sentiment.
Australia's S&P/ASX 200 index rose 0.22%, while New Zealand's S&P/NZX 50 index edged down 0.27% after rising in early trade.
Asian markets have been buoyed by positive signs around China's economic recovery, although the coronavirus pandemic continues to wreak economic havoc globally and raise concern about high valuations.
Investors will remain cautious ahead of the first U.S. presidential debate later in the day (Wednesday 0100 GMT), and as lawmakers continue efforts to cobble together additional economic stimulus.
U.S. consumer confidence and home price data is also due later. Upcoming U.S. economic data should help show how well the country is positioned to rebound from pandemic lockdowns, and how necessary more stimulus will be.
"Globally, a loss of momentum and the renewed rise in COVID-19 infection rates points to the need for additional fiscal and monetary support. That policy outlook is continuing to provide a supportive backdrop to equities despite recent volatility," ANZ Bank analysts wrote in a note.
U.S. House of Representatives Speaker Nancy Pelosi said on Monday that Democratic lawmakers unveiled a new, $2.2 trillion coronavirus relief bill, which she said was a compromise measure that reduces the costs of the economic aid.
U.S. traders posted strong gains on Wall Street on Monday, particularly in hard-hit sectors like hotels, banks and airlines which posted sizeable gains after several days of decline.
On Wall Street, Dow Jones Industrial Average rose 1.51%, the S&P 500 gained 1.61%, and the Nasdaq Composite was up 1.87%.
But there were still some signs of caution, as Europe is experiencing a rise in new COVID-19 infections and some U.S. states continue to grapple with high case numbers.
Safe-haven spot gold added 0.21% to $1,884.77 an ounce. U.S. gold futures gained 0.54% to $1,883 an ounce.
U.S. Brent crude slipped 19 cents to $42.24 a barrel while U.S. light crude was down 17 cents at $40.43 on demand worries.
The U.S. dollar dropped from a two-month high against a basket of currencies Monday, with the dollar index falling 0.3%, its biggest daily percentage drop in roughly three weeks.
Bonds were broadly steady. The yield on benchmark 10-year U.S. government debt fell half a basis point to 0.6577%.
(Reporting by Kane Wu in Hong Kong and Pete Schroeder in New York; editing by Richard Pullin) ((Pete.Schroeder@thomsonreuters.com; 202-310-5485;))