MSCI's broadest index of Asia-Pacific shares outside Japan dipped 0.1 percent in early trade, led by fall in Australia, though activity is likely to be slow due to holidays in China including Hong Kong.
Japan's Nikkei gained 0.25 percent though the Topi, broader and better gauge of the market, was down 0.2 percent.
Casting a shadow in Asia, two surveys showed on Sunday that growth in China's manufacturing sector sputtered in September as domestic and export demand softened..
The private Caixin/Markit factory Purchasing Managers' Index (PMI) fell to 50.0, the lowest reading since May 2017, in a stark reminder that the trade disputes are starting to have real consequences on the economy.
"The escalation of trade tensions between the U.S. and China recently has likely weighed on purchasing managers' sentiment as reflected by softer readings in trade-related sub-indices," wrote economists at Bank of America Merrill Lynch.
"Deterioration of PMI in September is in line with our expectation that coincident growth indicators will get worse before getting better. As U.S.-China trade conflict intensifies, we expect China's policymakers to step up monetary and fiscal easing to soften the blow from higher tariffs," they added.
The world's share markets had rallied in September on hopes that China and the United States will eventually work out a deal on trade, but talks proposed for around the end of the month collapsed after the two sides launched more tit-for-tat tariffs.
MSCI's broadest gauge of the world's shares covering 47 markets <.MIWD00000PUS> rose for the third straight month in September.
In the currency market, the Canadian dollar rose about 0.5 percent to a four-month high of C$1.2838 to the U.S. dollar
Canadian and U.S. negotiators "made lots of progress" on Sunday on a renewed NAFTA, but had still not settled tough issues such as American tariffs and access to Canada's dairy market, an official and sources said.
U.S. President Donald Trump's administration has said Canada must sign on to the text of the updated North American Free Trade Agreement by midnight on Sunday (0400 GMT Monday) or face exclusion from the pact.
Canadian media said Prime Minister Justin Trudeau will hold a cabinet meeting at 0200 GMT.
On the other hand, the euro was hit by worries about rise in Italy's fiscal deficit after the Italian government agreed to set a higher than expected budget deficit target that could put Rome on a collision course with Brussels.
The common currency traded at $1.1608
Oil prices stayed near recent peaks, with international benchmark Brent trading near a four-year high, as U.S. sanctions on Tehran squeezed Iranian crude exports, tightening supply even as other key exporters increased production.
Brent crude futures
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