The European Central Bank and the Federal Reserve (FED) are expected to cut interest rates to boost global stock markets, while the pound is weakened by fears that Johnson, who may become the new prime minister, will lead the UK out of Europe without an agreement.
The MSCI Mingsheng Asia-Pacific (excluding Japan) index rose 0.15%.
Nikkei stock index. N225 rose 0.95%.
China's Shanghai Composite Index. SSEC rose 0.2%. The Australian stock market rose 0.4% and the Korean Composite Stock Index. KS11 rose 0.45%.
The overnight S&P 500 index. SPX rose to a record high, supported by expectations that the Federal Reserve will cut interest rates at its policy meeting on July 30-31.
European stock markets also rose slightly on Monday, with the European Central Bank expected to cut interest rates by 10 basis points on Thursday.
But because the central bank's relaxation policy is no longer a new subject, market growth is limited.
"The possibility of a Fed rate cut supports the stock market, but the market has basically absorbed the expectation of a 25-basis-point cut," said Soichiro Monji, senior strategist at Sumitomo Mitsui DS Asset Management.
On the currency side, the pound fell 0.1% to $1.2465 against the dollar GBP = D4 and is expected to fall for the third consecutive day.
The pound is under pressure because Britain's ruling Conservative Party may elect Eurosceptic Johnson as the new leader and next prime minister to replace Teresa May. The results of weeks of intra-Party elections will be announced on Tuesday.
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