Moves were modest, with MSCI's broadest index of Asia-Pacific shares outside Japan dipping 0.4 percent in thin trade, with Chinese blue chips off 0.7 percent.

Going the other way, Japan's Nikkei firmed 0.3 percent and South Korea stocks rose 0.5 percent.

EUROSTOXX 50 futures also edged up 0.2 percent, steadying after Monday's retreat, while FTSE futures rose 0.1 percent.

The euro, French stocks and government debt had been shaken after investors assessed whether the right wing can repeat their success in French elections and how much sway far-right parties can have on the new European Union executive.

Bond yields rose across Europe, with the spread between French and German debt widening notably, after an opinion poll suggested the far-right National Rally could win the snap election, albeit without a clear majority.

Elsewhere, markets gave a muted reaction to Apple's long-awaited AI strategy, which integrates "Apple Intelligence" technology across a suite of apps. The iPhone maker's shares were down 0.3 percent in after-hours trade, having slipped 1.9 percent in normal hours.

S&P 500 futures and Nasdaq futures both eased 0.1 percent in Asian trading, after edging higher on Monday.

The market has, so far, proven remarkably resilient to the jump in US yields that followed Friday's jobs report and the pull back in expectations for Federal Reserve rate cuts.

"We see diminished prospects for easing this year, and now expect the first Fed cut only in November," analysts at JPMorgan said.

"Equities seem to be ignoring the plethora of risks, including politics, geopolitics, the narrow market concentration and the surge in meme stock and crypto trading that may signal froth," they added. "As such, we maintain a defensive tilt in our model portfolio."

Futures imply 37 basis points of Fed easing for this year, compared to 50 bps before the jobs report.

The Fed is considered certain to hold steady at its policy meeting on Wednesday, with the focus on whether it keeps three rate cuts in its "dot plot" projections for this year.

"We expect the dots to show two cuts in 2024, four cuts in 2025, three cuts in 2026 and a slight tick up in the longer-run or neutral rate," said analysts at Goldman Sachs in a note.

"We think the leadership would prefer a two-cut baseline to retain flexibility, but a one-cut baseline is a possible risk, especially if core CPI surprises to the upside on Wednesday."

The consumer price index (CPI) is forecast to rise a slim 0.1 percent in May, but with the core up 0.3 percent.

In currency markets, the euro steadied around $1.0766, after hitting a one-month low overnight at $1.0733. It has lost about 1.1 percent in the past two sessions, undermined by the US jobs reports and political uncertainty.

The dollar was broadly supported at 157.17 yen and just short of its May top of 157.715.

The weakness of the yen is one reason the Bank of Japan (BoJ) might decide to taper its bond buying at a policy meeting on Friday, as a step toward another rate hike.

Gold was just above one-month lows at $2,306 an ounce, after getting whiplashed by the pullback in market pricing for US rate cuts.

Oil prices consolidated Monday's 3 percent rally, as various investment banks tipped strong summer demand for fuel and potential US crude purchases for its petroleum reserve.

Markets are also awaiting monthly oil supply and demand data from the US Energy Information Administration and OPEC on Tuesday, and the International Energy Agency on Wednesday.

Brent dipped 4 cents to $81.59 a barrel, while US crude was unchanged at $77.74 per barrel.