The likelihood of Greece leaving the euro has been growing steadily since early May, when political parties opposed to the austere terms of the country's financial rescue received unexpectedly high support.
Surveys over the weekend showed that Greeks, while angry after more than two years of austerity measures that have seen salaries and pensions cut and new taxes imposed, still prefer Greece to keep the euro currency and not revert back to the drachma.
Still, the May election results were so splintered that it left the country without a coalition government. Another election has been set for June 17.
"With less than three weeks to the Greek elections, uncertainty remains whether a pro-bailout coalition can be formed," analysts at Credit Agricole CIB in Hong Kong said in an email.
After a higher open, Japan's Nikkei 225 index fell marginally to 8,579.03.
Hong Kong's Hang Seng was down less than 0.1 per cent to 18,703.01. But Australia's S&P/ASX 200 rose 0.5 per cent to 4,050.60 as traders scooped up bargains among resource shares that have underwent recent selloffs.
South Korean markets were closed for a public holiday. Wall Street will be closed Monday for Memorial Day, which typically results in subdued stock trading globally.
Debt-mired Greece has been kept solvent since May 2010 through loans the European Union and the International Monetary Fund.
A first bailout package of 110 billion euros ($138 billion) was agreed in May 2010 and a second, 130 billion euros deal was approved in March 2012.
A separate deal with private creditors earlier this year allowed Greece to write off nearly 107 billion euros of the country's 368 billion euro debt.
Later in the week, the U.S. government will release employment data for May.
China will release data on manufacturing for May. A private survey last week showed activity weakened further in May.
Benchmark oil for July delivery was up 98 cents to $91.64 a barrel in electronic trading on the New York Mercantile Exchange. The contract rose 20 cents to settle at $90.86 in New York on Friday.
In currencies, the euro rose to $1.2589 from $1.2518 late Friday in New York. The dollar fell to 79.40 yen from 79.66 yen.