* Traders look ahead to EU leaders' Wednesday summit * Treasury auctions $35 bln of two-year notes * Benchmark yields remain near 60-year lows By Chris Reese NEW YORK, May 22 (Reuters) - Benchmark U.S. Treasury debt prices fell on Tuesday for a third consecutive session as investors took profits from recent gains and pushed for lower prices ahead of government debt auctions later this week. Treasury yields last week fell to levels near their lowest in at least 60 years as fears that Europe will face new funding stresses added to demand for safe-haven U.S. debt. While a solution to Europe's debt woes did not appear to be imminent, traders took the lack of big headlines early in the week as an opportunity to cash in on the lofty prices while waiting to see how European leaders will address the crisis. "There was clearly a large unwind of risk, and it feels like it has kind of run its course," said Scott Graham, head of government bond trading at BMO Capital Markets in New York, adding: "I suspect we will bounce back and forth here, though if there is going to be substantial progress made in Europe, then we are way too expensive." Traders are also preparing for supply of $35 billion of five-year notes on Wednesday and $29 billion of seven-year notes on Thursday after a $35 billion sale of two-year notes on Tuesday. Investors often try to reduce prices heading into such auctions. The two-year note sale was met with near-average demand and Treasury debt prices traded steady at lower levels following the auction, with benchmark 10-year notes down 10/32 in price to yield 1.78 percent, up from 1.74 percent on Monday. Despite a third straight session of higher yields, the rate remains only marginally above the 1.67 percent level touched last September, which marked the lowest in at least 60 years. Traders will focus on an informal summit of EU leaders in Brussels on Wednesday, where French President Francois Hollande will push a proposal to move to bonds that are jointly underwritten by all euro zone member states. Germany opposes any such move, saying more progress is needed on coordinating fiscal policies across the euro zone. The monthly U.S. employment report scheduled for next week and Greek elections next month will help determine further market moves and whether yields retest recent lows. Safe-haven demand for Treasuries was also undermined on Tuesday by data showing U.S. home resales rose in April to their highest annual rate in nearly two years, while a falloff in foreclosures pushed home prices higher. "While the rise in sales is much flatter than in previous recoveries, with longstanding structural issues including tepid job creation and limited income growth holding it back, some sidelined demand is slowly returning to the market," said Lindsey Piegza, economist at FTN Financial in New York, adding "housing appears on a far more solid footing than a year ago." Price losses were pared on Tuesday afternoon after Dow Jones quoted former Greek prime minister Lucas Papademos as saying that Greeks had no choice but to stick with a painful austerity program or face a damaging exit from the euro zone, a risk he said was unlikely to materialize but was real. Papademos also told the news agency that some European states and institutions were considering contingency plans for any eventuality. Thirty-year bonds traded 1-9/32 lower in price to yield 2.87 percent, up from 2.81 percent late Monday.