Chinese markets were downbeat in morning trade, but Tokyo's key Nikkei index and stocks in Sydney and Singapore were up following a slightly higher close on Wall Street.
US President Joe Biden and European officials had sought to calm investors as bank shares tumbled on Friday, triggered by concerns over troubled lender Deutsche Bank.
German Chancellor Olaf Scholz assured traders that Deutsche Bank is "very profitable" after its shares nosedived by as much as 14 percent before closing 8.5 percent lower.
The German bank returned to financial health last year following a major restructuring after years of problems.
After markets closed on Friday, Biden said banks are "in pretty good shape" following the recent financial sector turmoil.
Clifford Bennett, chief economist at ACY Securities, said Monday that it was unlikely the German government would allow Deutsche Bank to collapse or face restructuring.
But the situation shows "the continuing and growing pressure on the banking system among the major Western economies", he wrote in a note.
"No bank is immune in the current climate. The forces that lead to the crisis so far seen, of higher rates and depositor uncertainty, only continue to grow."
Hong Kong was nearly two percent lower in Monday morning trade, with Shanghai down 0.7 percent. Taipei and Seoul both lost 0.5 percent, and Jakarta was down 0.4 percent.
But Tokyo was up 0.3 percent at the break, Singapore rose one percent and Sydney gained 0.2 percent.
Markets had rallied last week after financial authorities took steps aimed at preventing contagion from the collapse of US regional lenders this month.
But sentiment soured following decisions by central banks in the United States, Britain and Switzerland to hike interest rates, despite concerns about the impact of the monetary tightening on banks.
Amir Anvarzadeh of Asymmetric Advisors said markets would "remain in a state of flux as concerns about the health of the global banking system persist".
At the same time, "the market seems to have come to the view that the latest banking turmoil will do much of the work in taming inflation and chances for easier monetary policy this year have dramatically increased", he said.
- Key figures around 0230 GMT -
Tokyo - Nikkei 225: UP 0.31 percent at 27,471.17 (break)
Hong Kong - Hang Seng Index: DOWN 1.96 percent at 19,525.08
Shanghai - Composite: DOWN 0.74 percent at 3,241.15
Euro/dollar: UP at $1.0765 from $1.0764 on Friday
Pound/dollar: UP at $1.2238 from $1.2230
Euro/pound: DOWN at 87.95 pence from 87.96 pence
Dollar/yen: UP at 130.75 yen from 130.70 yen
West Texas Intermediate: FLAT at $69.24 per barrel
Brent North Sea crude: DOWN 0.1 percent at $74.94 per barrel
New York - Dow: UP 0.4 percent at 32,237.53 (close)
London - FTSE 100: DOWN 1.3 percent at 7,405.45 (close)
kaf/qan
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