* 4-week US bill auction garners lowest rate since early April * Markets pricing small probability of ECB rate cut in June * Such bets likely to accumulate in coming days By Chris Reese and Marius Zaharia NEW YORK/LONDON, May 30 (Reuters) - The U.S. Treasury sold four-week bills at the lowest interest rate in nearly two months on Wednesday, as investors worried over the implications of Europe's debt crisis continued to buy shorter-dated U.S. debt as a safe-haven. The Treasury sold the $30 billion of four-week bills at a high rate of 0.06 percent, the lowest since a similar auction of the bills on April 3 brought a high rate of 0.055 percent. Still, Wednesday's auction rate trumped the four-week auctions in December and January, when a series of sales brought a high rate of zero. Separately, a Treasury sale of $25 billion of one-year bills on Wednesday brought a high rate of 0.185 percent, unchanged from the two previous weekly auctions. Three-month dollar-denominated London Interbank Offered Rates (Libor) fixed on Wednesday at 0.46685 percent, unchanged from Tuesday. Three-month dollar Libor has held relatively steady since mid-April, after falling from a recent high near 0.58 percent in early January. Meanwhile, bets that the ECB will cut interest rates next week have reappeared in money markets, as Spanish and Italian debt yields approach levels that caused the central bank to introduce unprecedented easing measures last year. The threat that Greece could eventually leave the euro and worries over Spain's banking sector have prompted investors to sell Spanish and Italian debt, bringing the two countries' borrowing costs closer to levels deemed as unsustainable. The sheer size of their debt markets and their deep-rooted connections with other financial systems in the euro zone are reasons for investors to speculate that a policy response is being contemplated. The European Central Bank is seen as the most likely institution to take such measures to cool market jitters because it can act faster than politicians. It has done it before in the past by injecting around 1 trillion euros of cheap loans into financial system in December and February. Euro zone economic data this month has also been poor, supporting expectations that the ECB may soon resume monetary easing, possibly by cutting its key refinancing rate by 25 basis points from a record low of 1 percent. "Data ... have been softer, and then you have the Greece issue continuing to be unresolved and the Spanish issue continuing to be unresolved," said Elaine Lin, a rate strategist at Morgan Stanley, whose economists predict a rate cut. She said the euro overnight Eonia rate forward market was only pricing an over 10 percent probability of a rate cut in June and the chances were higher by another 10-20 percentage points for the July meeting. However, she expected markets to factor in a higher probability in the next few days. A key rate cut, if also accompanied by a cut in the 25 basis points deposit facility rate, could trigger a 5-10 bps fall in the near-term forward Eonia rates toward the 20 bps level seen now in September-October Eonia forward rates, Lin said. The lowest point on the 2012 Eonia curve is December, at 16 basis points, implying an 80 percent probability that the deposit rate would be slashed in half, according to BNP Paribas rate strategist Matteo Regesta. A Reuters poll of economists showed the ECB was likely to resist pressure to cut interest rates in June, but also pointed to a growing probability that it will reduce them later this year. Speculation about ECB monetary easing has also fueled a rally in Euribor futures , implying bets for lower fixings of benchmark euro zone interbank three-month Euribor rates later this year. The December Euribor future has gained back most of its losses made since Greece's inconclusive election on May 6, that sparked fears the country may be exiting the bloc. The fall earlier this month also coincided with unwinding bets that the ECB would have cut rates in May.