In early European trade, pan-region Euro Stoxx 50 futures rose 0.46% while German DAX futures climbed 0.57% and FTSE futures 0.40%.
In Asia, Japan's Nikkei rose 0.95% while MSCI's broadest index of Asia-Pacific shares outside Japan gained 0.04%.
The Shanghai Composite Index edged up 0.15%. Australian stocks added 0.4% and South Korea's KOSPI gained 0.45%.
The S&P 500 edged up towards a record high on Monday, supported by expectations that the Federal Reserve would cut interest rates at its July 30-31 policy meeting.
European stocks had also nudged higher the day before with the European Central Bank seen cutting rates by 10 basis points at its policy meeting on Thursday.
But with central bank easing no longer a fresh theme, market gains were limited.
"The likelihood of easing by the Fed is supportive for equity markets, but the probability of a 25 basis point rate cut has already been factored in for the most part," said Soichiro Monji, senior strategist at Sumitomo Mitsui DS Asset Management.
In currency markets, the pound was 0.2% lower at $1.2449 and headed for its third session of losses.
Sterling was under pressure due to the likelihood that Britain's ruling Conservative party would elect eurosceptic Johnson as its new leader and prime minister, replacing Theresa May. The result of the weeks-long internal party election will be announced on Tuesday.
Some investors are worried Johnson could pull Britain out of the European Union on Oct. 31 without a trade deal in place in order to appease hardline anti-EU members of his Conservative Party.
The dollar index against a basket of six major currencies rose 0.2% to 97.463, helped by a rise in U.S. Treasury yields.
The greenback gained 0.15% to 108.040 yen.
The euro dipped 0.17% to $1.1189, weighed by the possibility of easing by the ECB.
"It is going to take a bold stroke by the ECB to both satisfy markets clamouring for incremental easing and make a difference to the economy, all the while remaining inside its institutional setting and not destabilising the financial system," wrote Carl Weinberg, chief international economist at High Frequency Economics.
The New Zealand dollar slipped 0.4% to $0.6731, pressured in part by news the Reserve Bank of New Zealand (RBNZ) was taking a fresh look at unconventional monetary policy strategies, with interest rates already at a record low of 1.5%.
Crude oil prices edged higher after two days of sharp gains due to heightened tensions in the Middle East.
Brent crude added 0.19% to $63.38 per barrel after rising 1.2% the previous day on concerns over possible supply disruptions after Iran's seizure of a British tanker late last week.