Japan's benchmark Nikkei 225 gained nearly 0.3% in early trading to 22,548.07. South Korea's Kospi picked up 0.2% to 2,065.68, while Hong Kong's Hang Seng added 0.2% to 26,778.99. The S&P/ASX 200 in Australia lost 0.1% to 6,640.40, while the Shanghai Composite slipped 0.1% to 2,934.30.
Shares fell in Taiwan and were mixed in Southeast Asia.
British Prime Minister Boris Johnson is trying to win over rebellious lawmakers in time to meet the Oct. 31 Brexit deadline for the UK's exit from the 28-nation European Union.
A vote over the weekend ended with an amendment that delays the proposed deal, leaving the situation uncertain. And EU officials have not yet responded to Johnson's reluctant request for an extension of the month's end deadline.
"The can is not kicked far down the road with UK Prime Minister Boris Johnson expected to seek a new 'meaningful vote' on his deal as soon as Monday with the countdown to the Brexit deadline," Jingyi Pan of IG said in a commentary.
Meanwhile, Japan reported that its exports fell 5.2% from a year earlier in September while imports slipped 1.5%. The resulting deficit of 123 billion yen ($1.1 billion) reflected weak exports to China, South Korea and other Asian countries, customs data showed.
The mixed performance to start the week is a continuation of the wobbles that ended last week, when the S&P 500 index logged its second straight weekly gain even though stock indexes ended lower on Friday.
Technology companies led the slide, which erased the major U.S. indexes' gains from the day before. Communication services, industrials and health care stocks also fell, outweighing gains in real estate companies, banks and elsewhere in the market.
Investors are focusing on company earnings reports, searching for a clearer picture on the impact that the trade war between the U.S. and China is having on corporate profits and the broader economy.
The S&P 500 index fell 0.4% to 2,986.20. The index is just 1.3% below its all-time high set in late July.
The Dow Jones Industrial Average dropped 1% to 26,770.20 and the Nasdaq lost 0.8%, to 8,089.54. The Russell 2000 index of smaller stocks gave up 0.4% to 1,535.48.
Uncertainty over the standoff between Beijing and Washington has been roiling markets. Negotiators reached a truce last week that kept the conflict over trade and technology from escalating further, but both sides still have many issues to work out before reaching a substantive deal.
ENERGY: Benchmark crude oil dipped 10 cents to $53.68 a barrel in electronic trading on the New York Mercantile Exchange. It fell 15 cents to $53.78 a barrel Friday. Brent crude oil, the international standard, dropped 20 cents to $59.22 a barrel.
CURRENCIES: The dollar rose to 108.50 Japanese yen from 108.38 yen on Friday. The euro slipped to $1.1158 from $1.1174.
The summary of this story has been corrected to show Brexit deadline is Oct. 31, not 21.
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