Japan's Nikkei 225 index fell 0.7 percent to 8,781.92. South Korea's Kospi was 0.7 percent lower at 1,951.23 and Hong Kong's Hang Seng dropped 0.5 percent to 10,394.33. Australia's S&P/ASX 200 lost 0.3 percent at 4,274.70.
Benchmarks in Singapore and the Philippines also fell. Shares in mainland China were mixed after being closed for a week for Chinese New Year holidays. Taiwan and New Zealand rose.
European leaders were to meet later Monday in Brussels to discuss austerity and belt-tightening measures as well as a tentative deal reached Saturday between Greece and its private investors that could avert a disastrous Greek default on its debt.
If the deal holds and works, it will help prevent a potential shock to the world banking system. But it doesn't resolve the weakening economic conditions in Greece and other European nations as they rein in spending to get their debts under control.
Stan Shamu of IG Markets in Melbourne said that "the Greece debt issues will remain a source of uncertainty and might dampen the risk mood ahead of the EU summit today."
Under the agreement, investors holding 206 billion euros ($272 billion) in Greek bonds would exchange them for bonds with half the face value. The replacement bonds would have a longer maturity and pay a lower interest rate.
The deal would reduce Greece's annual interest expense from about 10 billion euros to about 4 billion euros. When the bonds mature, Greece would have to pay its bondholders only 103 billion euro.
It is unclear how investors who buy and sell the bonds of other debt-burdened countries, such as Italy, Spain and Portugal, will react. If they drive up borrowing costs for those countries, the debt crisis could get worse.
Private investors hold two-thirds of Greece's debt, which is equal to an unsustainable 160 percent of its annual economic output. By restructuring the debt, Greece hopes to make it a more manageable 120 percent by decade's end.
On Wall Street, stocks mostly fell Friday after the government said the US economy grew more slowly than expected in the last three months of 2011.
Economic growth for October through December came in at an annual rate of 2.8 percent. That was the fastest of 2011 but lower than the 3 percent that economists were looking for.
The Dow Jones industrial average fell 0.6 percent to 12,660.46. The Standard & Poor's 500 index fell 0.2 percent to 1,316.33. The Nasdaq composite rose 0.4 percent to 2,816.55.
Benchmark oil for March delivery was down 36 cents to $99.20 per barrel in electronic trading on the New York Mercantile Exchange. The contract fell 14 cents to end at $99.56 per barrel on the Nymex on Friday.