The pan-European STOXX 600 was up 0.2% by 08:02 GMT, after jumping 1.3% in the previous session to its highest level since February 2020.
The benchmark eurozone stock index jumped by 0.1% while MSCI's broadest index of Asia-Pacific shares outside Japan advanced 0.4%.
Europe's STOXX 600 rose in early trading, up 0.1% at 0843 GMT, Germany's DAX was up 0.4%, but London's FTSE 100 was down 0.7%.
The pan-European STOXX 600 index was down 0.7% by 08:10 GMT while benchmarks rose in Japan but fell in South Korea, Australia and China.
Shanghai, Tokyo, Hong Kong and Seoul all advanced while the pan-European STOXX 600 index was up 0.4% by 08:47 GMT.
The benchmark European stock index was down 0.6%, London's FTSE 100 slipped 0.2% and Germany's DAX lost 0.1%, both well off session lows.
Asian stock markets were mostly higher Tuesday after Wall Street rose on a flurry of corporate deals and China's economic activity improved. Shanghai, Hong Kong and Seoul gained, while Tokyo retreated.
Most Asian markets fell Monday following another disappointing performance on Wall Street with investors growing concerned about an uptick in coronavirus infections in Europe and the United States, as well as the lack of movement in Washington on a new stimulus.
Most major markets across Asia were in the green with MSCI's broadest index of Asia-Pacific shares outside Japan rising 0.9% for its second straight session of gains.
The MSCI index is also set for its biggest weekly drop since March, down more than 4% so far this week. Chinese blue-chips dropped 1.6%, Hong Kong's Hang Seng fell 1.7%, Seoul's KOSPI sank 2.59% and Australian shares fell 0.81%.
Japan's Nikkei declined 1.11%.