(Updates throughout) * European stock index turns negative * Bond yields in Europe, U.S. steady * Oil prices gain for a third session in a row * Currencies moves limited to tight ranges * Graphic: World FX rates tmsnrt.rs/2RBWI5E LONDON/SYDNEY, June 3 (Reuters) - World stocks clung close to record highs on Thursday as investors weighed inflation concerns ahead of key U.S. economic data, while oil prices rose for a third straight session. Market sentiment was cagey as investors backed away from big bets before the release on Friday of U.S. jobs data, which should offer further clarity on whether the faster-than-expected pace of economic recovery can be sustained and what that might mean for monetary policy. In Europe, the broad Euro STOXX index was 0.2% down, drifting away from record highs scaled on Tuesday, with Britain’s FTSE 100 slipping 0.7%, while Germany’s DAX and the French CAC 40 were both down 0.2%. There was a similar pullback in Asia, with MSCI’s broadest index of Asia-Pacific shares outside Japan shedding 0.2% after reaching three-month highs on Wednesday. In Japan, the Nikkei share average rose 0.4%, while Australian shares climbed to all-time highs as investors cheered stronger-than-expected economic growth data released on Wednesday. The MSCI world equity index, which tracks shares in 49 countries, hovered in and out of positive territory, below Tuesday’s record high. U.S. futures pointed to a dip on Wall Street at the open . 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 the monthly jobs numbers on Friday. Investors will be 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. European Central Bank (ECB) chief Christine Lagarde said on Wednesday the ECB will support the euro zone “well into” its recovery from a pandemic-induced double dip recession. Those comments helped euro zone bond yields hold near record lows on Thursday. Germany’s 10-year yield, the benchmark for the bloc, was up less than a basis at -0.19%. In the U.S., 10-year Treasury yield were also steady ahead of the U.S. economic data release. 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, rose to 90.112, having found strong support around the 89.946 mark in recent sessions after falling 2% in April and a further 1.6% in May. Against the euro the dollar traded 0.2% higher at $1.2183 and it crept a fraction higher on Antipodean currencies. Brent rose 0.4% to $71.62 a barrel, after earlier reaching the highest since September 2019. U.S. crude futures went as high as $69.40, the highest since October 2018. Spot gold was down 0.1% to $1,892.26 per ounce. Editing by Sam Holmes and Toby Chopra Our Standards: The Thomson Reuters Trust Principles.