* Spot Eonia falls to record low
* ECB rate cut, liquidity injection still to take effect
* Forward breakevens rise with inflation outlook
* Greek yields drop, Cyprus plans comeback (Adds fresh quotes, updates prices)
By John Geddie
LONDON, June 10 (Reuters) - Euro zone overnight interbank interest rates traded at record lows on Tuesday in a sign that the European Central Bank's efforts to keep money markets anchored will bear fruit.
Strategists said markets were adjusting to last week's cut in the ECB's main interest rates and its promise of fresh liquidity for banks, which should see rates fall even lower when these measures come into effect this week and next.
"The ECB has given very strong forward guidance for the first two years, and all its measures work to pin front-end rates," said Michael Michaelides, rates strategist at RBS.
Spot Eonia fixed at 0.053 percent after markets closed on Monday, dropping below the previous historic low set in February 2013.
Some argue, however, that the move could be short-lived, with low trading volumes during European holidays on Monday creating a slight anomaly.
"The drop could be psychological but I don't find that very satisfactory because the new rates do not yet apply," said Christoph Rieger, a strategist at Commerzbank.
"We have observed in the past that during German holidays when volumes fell, Eonia fixings also declined."
Rieger expects Eonia to fix higher on Tuesday and start to fall after the European Central Bank implements cuts in its main rates on Wednesday, imposing for the first time charges on cash parked by banks at the ECB.
The excess liquidity in the banking system will also keep downward pressure on short-term rates.
Excess liquidity stands at 120 billion euros, well above the three-year low of 70 billion euros hit at the end of last month, although the ECB took a further 25 billion euros from its cash offerings on Tuesday.
From next week, the ECB will inject around 170 billion euros into the banking system by halting tenders that withdrew funds spent on past government bond purchases.
It has also introduced 400 billion euro ($544.86 billion) of ultra-cheap four-year loans for banks - conditional on their lending to the smaller companies that are Europe's economic backbone - which will be available from September.
European Central Bank Governing Council member Erkki Liikanen reiterated on Tuesday that the ECB still has tools it can employ if needed.
Forward Eonia rates are already pricing in the ECB's ability to keep the market pinned down, while six-month contracts dated for 12-months time are low enough to suggest further policy loosening, say strategists.
The fall in Eonia rates was partly behind a fall in the euro against the dollar and yen on Tuesday, strategists said.
Market confidence in the ECB's ability to nurture growth and reflate the economy has been evident in an uptick in forward breakevens over the last few days.
Expectations of an eventual rise in consumer prices should bode well for France which hired a group of banks to sell a new 15-year inflation-linked bond on Tuesday.
In German government bonds - a benchmark for euro zone borrowing - these expectations can also be shown by investors' preference for short-dated benchmarks over longer maturities, which led to further curve steepening on Tuesday.
The ECB's pledge to keep rates at historic lows for some time has, however, fuelled demand for the higher-yielding bonds of some of the bloc's weakest links.
Yields on 10-year Greek bonds dropped 24 bps to 5.53 percent on Tuesday, a level not seen since January 2010, while Portuguese equivalents were within a whisker of euro era lows after dropping 14 bps to 3.24 percent.
These new lows raise the prospect that Greece, which returned to markets in April for the first time since 2010, could soon issue more debt to help to stave off the need for a third bailout, Commerzbank said in a note on Tuesday.
The Republic of Cyprus - bailed out just last year - is set to meet investors ahead of a possible euro bond, IFR reported on Tuesday.
Elsewhere, Italy's 10-year yields rose 2 bps to 2.72 percent from Monday's record low while Spain's dropped 3 bps to a new low of 2.56 percent. (Editing by Catherine Evans and Nigel Stephenson)