MSCI's broadest index of Asia-Pacific shares outside Japan was little changed, staying just under Friday's peak and two percentage point below its 2011 peak, having risen about 7% so far this year.
Japan's Nikkei average was little changed after closing at a 5-1/2-month high on Friday.
MSCI's All World share index, which tracks 45 countries, set its fourth consecutive record high on Friday, thanks to strong US economic data.
"The US job data was undeniably strong, confirming strong recovery in the economy. While some people say the valuations of US shares are becoming a bit expensive, I think there's further upside potential given US interest rates are still low," said Tsuyoshi Shimizu, chief strategist at Mizuho Asset Management.
Despite the improvement in the job market, the Federal Reserve is widely expected to keep interest rates low for at least a year even as it is trimming stimulus.
In addition, two other major central banks -- the European Central Bank and the Bank of Japan -- are committed to stimulus, keeping cheap money sloshing around possibly for years.
Investors are now looking to quarterly corporate earnings in the United States and elsewhere, with Alcoa kicking off the US earnings seasons on Tuesday.
"People said the US earnings would be bad for January-March but in the end the profits were up. I would expect decent results," Shimizu said.
In the currency market, the dollar kept its momentum after data on Thursday showed US employment growth smashed forecasts and unemployment fell to near a six-year low of 6.1%.
The dollar index held at 80.305, having recovered from two-month low of 79.740 hit on Tuesday.
The euro was on the back foot at $1.35875 while the yen traded at 102.10 yen to the dollar.
The US debt yield also rose, with the two-year yield near 10-month high around 0.532% on Thursday and last stood at 0.528%
Copper slipped from a 4-1/2-month high of $7,190 per tonne hit on Friday but still held near that level at $7,154, thanks to increasingly positive economic signals in China, which will start releasing June data later this week.
Oil prices traded near one-month lows, partly on the prospects of revived Libyan exports. US crude futures traded at $103.81 per barrel, near a low of $103.67 hit late last week.