* Global shares hit highest level since September * Wall St extends gains on 'fiscal cliff' hopes * Euro hits fresh 7-1/2 month high vs U.S. dollar By Angela Moon NEW YORK, Dec 18 (Reuters) - Global shares hit their highest level since September on Tuesday on signs of compromise in U.S. talks to stop automatic tax hikes and spending cuts that could hurt the economy next year. As investors' appetite for safe-haven bonds and the dollar weakened, the euro climbed to a 7-1/2-month peak against the greenback. U.S. stocks extended sharp gains after Republican House Speaker John Boehner said he has hope for a broad deal on budget talks and also said he would begin work on income tax rate legislation while negotiations with the White House continue. He edged closer to President Barack Obama's position by proposing higher taxes on those who earn $1 million or more and extending lower tax rates for everyone else. However, the White House rejected Boehner's backup plan, saying it did not put enough of a tax burden on the wealthiest Americans. Obama had made a counter-offer to Republicans on Monday night that included a major change in his position on tax hikes for the wealthy, according to a source familiar with the talks. "As you get more and more clarity and dialogue that there will be a compromise to avoid a 'fiscal cliff' I think the markets are going to rally," said Weston Boone, vice president of listed trading at Stifel Nicolaus Capital Markets. "What's holding this market back - the S&P 500 - from continuing to reach higher highs is the macro headwinds, and a lot of that emanates from D.C." The Dow Jones industrial average was up 75.74 points, or 0.57 percent, at 13,311.13. The Standard & Poor's 500 Index was up 10.82 points, or 0.76 percent, at 1,441.18. The Nasdaq Composite Index was up 32.65 points, or 1.08 percent, at 3,043.25. European shares rose near 2012 highs on Tuesday, with the FTSEurofirst 300 index up 0.6 percent at 1,138.83. The rally pushed the MSCI index of global stocks up 0.8 percent, its highest level since September, with an 18-month peak also in sight. The euro hit a 7-1/2-month peak against the dollar above $1.32, boosted by optimism about the chances for a U.S. budget deal. A U.S. fiscal deal is viewed as positive for risk sentiment and should benefit riskier currencies such as the euro at the expense of the safe-haven dollar. The euro hit a high of $1.3201, its strongest since early May. It was last at $1.3198, up 0.3 percent. Brent crude rose $1.13 cents to $108.77 a barrel. U.S. crude oil gained 81 cents to $88.01 a barrel by midday. In early trade, it climbed above the 50-day moving average, a key technical indicator watched by traders. U.S. Treasury yields on Tuesday steadied just below their highest levels since October. The benchmark 10-year U.S. Treasury note was down 9/32, the yield at 1.8031 percent.