* European shares at fresh record highs * U.S. inflation numbers awaited * U.S. Treasury yields extend rising run * Euro-dollar near four-month lows * Gold, oil prices recover LONDON, Aug 10 (Reuters) - Global shares hovered below record highs on Tuesday, while anticipation of earlier tapering of bond-buying by the Federal Reserve pushed U.S. 10-year Treasury yields to their longest rising run in six months. The dollar also scaled a four-month high versus the euro as investors looked ahead to U.S. inflation numbers on Wednesday for further indications of when the world’s largest economy may start to withdraw stimulus after taper talk was amped up by strong jobs data. MSCI’s All Country World Index, which tracks shares across 49 countries, was 0.1% up on the day, below record highs reached last week. European shares pushed to fresh record highs, with the STOXX 600 0.2% stronger to extend its gains to a seventh straight session, boosted by travel and leisure companies. Nasdaq futures added 0.1% and S&P 500 futures were flat as investors awaited progress towards the passing of a much-anticipated infrastructure bill. The U.S. Senate has set a vote on passage of the $1 trillion bipartisan infrastructure bill for 11 am ET (1500 GMT), after which it would immediately begin to debate $3.5 trillion in additional investments. Activity, meanwhile, was heating up in bond markets. Indications in recent days of an improving labour market have prompted investors to rethink the outlook for U.S. monetary policy, halting recent sharp falls in both U.S. and European bond yields. U.S. 10-year Treasury yields scaled their highest in over three weeks, extending the longest run of gains since early February. The benchmark 10-year yield rose to as high as 1.336% in London trade and is up almost 20 basis points from six-month lows hit last week. “Risks remain. While inflation data has so far not been a major market mover, Wednesday’s July consumer price index release has the potential to cause volatility, especially given expectations that inflation has passed the peak,” said Mark Haefele, chief investment officer, UBS Global Wealth Management. Adding fuel to the debate, two Fed officials said on Monday that while the labour market still has room for improvement, inflation is already at a level that could satisfy one leg of a key test for the beginning of interest rate hikes. Data on Monday showed that U.S. job openings shot up to a fresh record high in June and hiring also increased. That followed Friday’s non-farm payroll report showing jobs increased by a larger-than-expected 943,000 in July. While signs of economic recovery in the U.S. are reviving reflation trade bets, investors remain wary of the lingering risks posed by COVID-19. China on Monday reported more COVID-19 infections in what seems to be its most severe resurgence of the disease since mid-2020, as some cities added rounds of mass testing in a bid to stamp out infections. MSCI’s broadest index of Asia-Pacific shares outside Japan added 0.4% after trading much of the day in the red as worries weighed about the spread of the Delta variant. With tapering expectations gaining traction, the dollar extended its gains made on Friday and Monday. The dollar index reached an 18-day high of 93.102 on Tuesday, while the euro hit a new four-month low against the dollar, with the pair changing hands at $1.1716. Gold prices recovered after touching a four-month low on Monday, with spot gold little changed at $1,729.80 per ounce. Oil prices rose more than 1% on Tuesday, recouping some of the losses in the previous session when prices hit a three-week low. U.S. crude oil futures were trading at $67.66 per barrel, up $1.17 or 1.76%. Brent crude was at $69.98, 1.36% higher. U.S. oil was up 1.7%, at $67.56 a barrel. Reporting by Tom Arnold in London and Paulina Duran in Sydney; Editing by Jacqueline Wong, Raissa Kasolowsky and Chizu Nomiyama Our Standards: The Thomson Reuters Trust Principles.