(Adds European, U.S. stock futures in para 2, analyst comment in paras 9-12, bullets; updates prices and levels throughout) * Asian stock markets : tmsnrt.rs/2zpUAr4 * European futures point to a record high for STOXX 600 * MSCI Asia ex-Japan slightly below Wednesday’s 3-month top * Oil prices gain for a third session in a row * Currencies moves limited to tight ranges SYDNEY, June 3 (Reuters) - Asian shares stepped back from three-month highs on Thursday as investors weighed inflation concerns ahead of key U.S. economic data while oil prices rose for a third straight session. Futures point to a firmer open for European stocks with the benchmark STOXX 600 set to flirt with a fresh record high. The pan-region Euro Stoxx 50 futures climbed 0.5% while futures for German DAX and London’s FTSE were up 0.1% each. U.S. stock futures, the S&P 500 e-minis , rose 0.3%. MSCI’s broadest index of Asia-Pacific shares outside Japan were up at 709.1 points, after reaching as high as 712.57 on Wednesday, a level not seen since early March. Japan’s Nikkei added 0.4%. Australian shares climbed to all-time highs as investors cheered stronger-than-expected economic growth data released on Wednesday. Chinese shares were marginally firmer in a choppy session. While broader stock markets remain close to record highs, the momentum seen earlier in the year has ebbed as investors worry a stronger-than-expected rebound from COVID-19 means higher inflation and sooner-than-expected monetary policy tightening. A weekly U.S. unemployment report and May private payrolls data on Thursday will be followed by monthly jobs numbers on Friday, with investors looking for signs of an economic rebound and rising inflation. Adding to inflation fears, oil prices hit the highest level since September 2019 on expectations for a surge in fuel demand later this year, particularly in the United States, Europe and China when major producers step up supply discipline. So far though, “increases in inflation expectations have coincided with equities performing well recently,” said Oliver Jones, senior markets analyst at Capital Economics. “In general, we suspect that these conditions will remain in place for a while longer.” Capital Economics forecasts that real global output will grow at the fastest rate in nearly 50 years this year. “While it is possible that major central banks eventually have to tighten policy faster than is widely expected if inflation does not fall back in the way they are anticipating, it will be hard to tell if this is happening until next year at the earliest,” Jones noted. Investment managers are also becoming increasingly worried with BlackRock Founder Larry Fink the latest to warn that the market was underestimating the risk of higher inflation. Philadelphia Fed Bank President Patrick Harker also restated his call that “it may be time to at least think about tapering our $120 billion in monthly Treasury bond and mortgage-backed securities purchases.” The Fed has already announced it would begin unwinding the corporate bond holdings it acquired last year to calm credit markets at the height of the pandemic. In Australia, the central bank is expected to begin tapering its pandemic emergency stimulus from next month when investors believe it would announce not extending its three-year yield target beyond the April 2024 bond. Wall Street’s main indexes ended Wednesday’s session mixed despite a breathtaking rally in theatre chain operator AMC Entertainment Holdings, which nearly doubled in price on Wednesday, lifting a group of stocks favoured by retail investors on forums such as Reddit’s WallStreetBets. Moves in currency markets have been limited with the dollar index and other major pairs staying in tight ranges. The dollar index, which measures the greenback against a basket of major currencies, was flat at 90.020, not far from a five month trough of 89.535 touched last week. The Japanese yen was barely changed at 109.74 per dollar. The Australian and New Zealand dollars were laggards, down 0.2% each at $0.7730 and $0.7215, respectively. Brent rose 49 cents to $71.84 a barrel, the highest since September 2019. U.S. crude futures went as high as $69.32, the highest since October 2018. U.S. West Texas Intermediate (WTI) crude rose 25 cents to $69.08 a barrel, its highest since October 2018. Editing by Sam Holmes Our Standards: The Thomson Reuters Trust Principles.