On Wall Street, futures for the Dow Jones industrial average and the broader Standard & Poor's 500 both were up 0.3 percent.
Traders regained confidence following concern about the soundness of Portugal's Espirito Santo International, which reportedly missed a debt payment last week, echoing issues that spawned Europe's debt crisis. Investors appeared to have been reassured any problems would be contained.
"Concerns about a widespread spill-over across Europe eased," said Mizuho Bank in a report. "Sentiment improved across the board."
Investors were looking ahead to announcements Tuesday about monetary policy in Australia and Japan and European inflation. U.S. Federal Reserve Chairwoman Janet Yellen is due to testify before a congressional committee.
On Wednesday, China is due to report quarterly economic growth. The country's top economic official, Premier Li Keqiang, said last week that the economy's performance improved compared with the first quarter, when growth tumbled to 7.4 percent, but gave no details.
"This week global markets will face U.S. earnings, central bank rhetoric and growth," said Chris Weston, chief strategist for IG, in a market commentary.
Tokyo's Nikkei 225 gained 0.9 percent to 15,296.82 and China's Shanghai Composite rose 1 percent to 2,066.65. Hong Kong's Hang Seng gained 0.5 percent to 23,346.67.
Seoul's Kospi gained 0.5 percent to 1,998.32 and Taiwan's Taiex added 0.5 percent to 9,545.99. Sydney's S&P ASX 200 rose 0.4 percent to 5,511.70.
Benchmark U.S. crude for August delivery fell 39 cents to $100.44 per barrel in electronic trading on the New York Mercantile Exchange. The contract plunged $2.10 to close at $100.83 in the previous session on expectations of higher supplies and muted global demand.
In currency markets, the dollar rose to 101.47 yen from 101.32 yen late Friday. The euro rose to $1.3628 from $1.3608.