Japan's Nikkei 225 index rose 0.2 percent at 9,177.92 while South Korea's Kospi fell by 0.2 percent to 1,942.82 and Hong Kong's Hang Seng shed 0.8 percent to 19,946.86.
In mainland China, the Shanghai Composite Index dropped 1 percent to 2,096.46.
Australia's S&P/ASX 200 was down less than 0.1 percent at 4,367.50. Taiwan's benchmark fell while New Zealand's rose.
Asian markets drifted lower, lacking direction in the absence of major economic data out this week in Asia and the summer lull.
The Shanghai benchmark hit the lowest level since early 2009 as investors continue to be cautious about the world's second largest economy amid mounting signs of a slowdown.
A report over the weekend that property prices in July rose slightly indicated the property market was benefiting from recent interest rate cuts, analysts said. But investors were still downbeat about prospects for further big stimulus measures, which are often announced on weekends or holidays.
"Every weekend investors are expecting the (Chinese) central government would throw out polices and every Monday comes in and there are no policies," said Jackson Wong, a vice president at Tanrich Securities. "That disappoints investors in general."
Investors this week are also watching Greek Prime Minister Antonis Samaras' visit to Germany and France, where he is expected to ask for an extension on Greece's deadline to meet fiscal targets as it carries out painful reforms.
A senior German politician said over the weekend that Greece doesn't have any wiggle room, darkening investors' mood over Europe's simmering debt crisis. There's also little chance of Germany's governing coalition supporting a third rescue package, Volker Kauder, a senior ally of Chancellor Angela Merkel, was quoted by Der Spiegel as saying.
Among individual stocks, Japan's embattled electronics maker Sharp Corp. plunged 6 percent. South Korea's Samsung Electronics fell 1.3 percent. Hong Kong-listed Lenovo Group lost 2.9 percent.
In the U.S. on Friday, stocks posted small gains on positive news from retailers. The Dow Jones industrial average rose 0.2 percent to close at 13,275.20. The broader Standard & Poor's 500 index rose 0.2 percent to close at 1,418.16. The Nasdaq rose 0.5 percent to close at 3,076.59.
Crude oil futures were up 34 cents to $96.35 per barrel in electronic trading on the New York Mercantile Exchange. The contract rose 41 cents to finish at $96.01 per barrel in New York on Friday.
In currencies, the euro rose to $1.2330 from $1.2322 late Friday in New York. The dollar fell to 79.46 yen from 79.55 yen.