Japan's Nikkei stock index edged down slightly, but was on track to log a rise of 0.8 percent in a week in which it touched its highest levels since August 2015.
Japan's Nikkei and South Korea's KOSPI were flat, while Australian shares rose0.4 percent.
The Nikkei 225 declined 0.26 percent and the Kospi fell 0.71 percent.
The Nikkei 225 gained 0.86 percent and South Korea's benchmark Kospi index rose 0.45 percent in early trade.
Markets in greater China were in the green, with Hong Kong's Hang Seng Index gaining 0.56 percent. On the mainland, the Shanghai Composite added 0.22 percent and the Shenzhen Composite rose 0.273 percent.
The Nikkei 225 edged lower by 0.6 percent while the Kospi gave up earlier gains to decline 0.75 percent.
MSCI's broadest index of Asia-Pacific shares outside Japan added 0.2 percent, while Japan's Nikkei put on 0.5 percent.
MSCI's broadest index of Asia-Pacific shares outside Japan rose 0.1 percent in early trade. Japan's Nikkei slipped 0.3 percent.
The Nikkei 225 was down by 0.64 percent and South Korea's benchmark Kospi index fell 0.75 percent.
MSCI's broadest index of Asia-Pacific shares outside Japan was down 0.27 percent while Japan's Nikkei rose 0.3 percent thanks to the dollar's rebound against the yen.
With U.K. elections, an European Central Bank policy meeting where policymakers may take a less dovish stance, and former FBI director James Comey's Senate testimony on all set for Thursday, market participants will be wary of taking big positions.
The S&P/ASX 200 tumbled 1.02 percent, driven by broad declines in the utilities, financials and energy sub-indexes. Major banking and resource stocks traded in the red.
The attackers first drove into pedestrians on London Bridge, then headed to Borough Market before being shot by police. At least seven were killed and 48 injured following the attack. Twelve individuals have already been arrested in the wake of the attack, which the Metropolitan Police said is being treating as a terrorist incident.
Data shows foreign investors, who make up 70 percent of trading activity in the Tokyo market, rushed to cover short positions as a rally from the year's low on April 17 gathered momentum.
MSCI's broadest index of Asia-Pacific shares outside Japan ticked up 0.1 percent, while Japan's Nikkei gained 1 percent to top the psychologically important 20,000-point level for the first time since August 2015.
Stocks in Asia were mostly flat on Thursday following a slightly lower close on Wall Street in the last session and ahead of the Caixin PMI data from China due later in the morning.
China's stock markets are in focus as they reopen after a long weekend with the official survey on manufacturing likely to set the tone for markets there and possibly the rest of the region. The PMI survey, due at 0100 GMT, is expected to show factory activity in China expanded at its slowest pace in eight months in May, according to a Reuters poll.
The Nikkei 225 edged 0.13 percent lower in early trade. The S&P/ASX 200 was effectively flat, trading 0.01 percent lower.
North Korea fired what looked to be a ballistic missile near its coast. The missile reportedly landed in the Sea of Japan after staying in the air for 6 minutes. This is the ninth missile the hermit state has tested this year, as it faces increasing pressure from the U.S. and historical ally China over its missile testing program.
The Nikkei 225 edged lower by 0.06 percent while the Kospi added 0.07 percent in early Friday trade. The Kospi is trading at an all-time high and in its sixth consecutive session of gains.
Japan's Nikkei dipped 0.1 percent though MSCI Japan rose 0.4 percent in dollar terms .
The Nikkei 225 jumped 0.79 percent in early Wednesday trade while the Kospi rose a more modest 0.31 percent. Australia's benchmark S&P/ASX 200 index edged higher by 0.1 percent.
The Nikkei 225 dipped 0.16 percent in Tuesday trade while the ASX 200 declined 0.05 percent. South Korea's benchmark Kospi index was up by 0.14 percent.
The Nikkei 225 was 0.34 percent higher. Australia's ASX 200 added 0.66 percent, driven largely by its energy and materials sub-indexes which were up 1.85 percent and 1.43 percent respectively.