* Italy, Spanish yields head towards record lows
* Bunds also rally on Fed stance
* Eonia rate fixes at record low due to ECB measures (Updates prices, adds more comments)
By Emelia Sithole-Matarise
LONDON, June 19 (Reuters) - Peripheral euro zone bond yields headed back towards historic lows on Thursday in a market-wide rally after the U.S. Federal Reserve signalled that rising inflation won't trigger an increase in rates any time soon.
Longer-dated euro zone bonds sold off earlier this week after higher-than-expected U.S. consumer price inflation raised the possibility that the Fed might be open to lifting interest rates sooner than many in the market had previously thought.
But the Fed did not note any inflation concerns after its policy meeting on Wednesday at which policymakers lowered their long-term rate target and affirmed a commitment to retaining accommodative monetary policy.
This accelerated the hunt for yield in euro zone bonds which were already buoyant from the European Central Bank's interest rate cuts and liquidity measures taken two weeks ago.
Italian and Spanish 10-year yields fell 8 basis points to 2.77 percent and 2.68 percent respectively, within sight of record lows hit last week, while Greek equivalents dropped 13 bps to 5.83 percent. Yields on top-rated euro zone bonds were 5-7 bps lower.
"The market was fearing a hawkish tone in the FOMC statement or at the press conference but the fact that (Fed Chair Janet) Yellen remained relatively accommodative is why the market is rallying," said BNP Paribas strategist Patrick Jacq.
German bonds, the euro zone benchmark, unwound some of their outperformance of U.S. Treasuries. The two-year yield gap in favour of Treasuries tightened by 9 bps to 34 bps.
The spread was still at its widest in seven years, reflecting the policy divergence between the Fed and the ECB. The Fed is trimming its monetary stimulus and is expected to start hiking rates next year while the ECB is set to maintain an ultra-easy policy for years.
Jacq at BNP Paribas said he expected Bunds to resume their outperformance of Treasuries, predicting the U.S. 10-year T-note yield premium would gain a further 10 bps in coming weeks to more than 130 bps, taking it back to 1999 peaks.
The ECB's June 5 decision to cut interest rates and inject liquidity into the market has also driven short-term money market rates to historic lows, with the overnight bank-to-bank Eonia rate tumbling close to zero.
Eonia fixed at 0.015 percent on Wednesday, the latest in a series of record lows after the ECB stopped a weekly deposit tender to neutralise the effect of the bond purchases it made at the height of the debt crisis, injecting tens of billions of euros into the market.
The central bank's measures have given extra impetus to the two-year long investor hunt for yield in the euro zone's weaker economies that has squeezed some borrowing costs to record lows.
This enabled Cyprus to return to markets on Wednesday a little more than a year after it was bailed out.
Greece also returned to market with an issue of five-year bonds in April. Athens plans another bond sale in coming months.
Investors' willingness to take higher risks to maximise profits is, however, exacerbating fears that some bonds have become over-valued and that some of the euro zone's struggling economies may ease off on fiscal reforms.
While he saw potential for peripheral euro zone yield premiums over German benchmarks to fall further, Amundi Asset Management's Philippe Ithurbide said it was "increasingly urgent to stick to what is liquid and demonstrably solvent".
"Not because we expect a resurgence in the banking or sovereign debt crisis, but simply because questions on excessive valuations will be increasingly legitimate," he said in a note.
Vanguard Group Inc's new bond chief, Gregory Davis, said at a Reuters summit on Wednesday that investors could be ignoring warning signs in less stable countries in their search for yield. His unease was echoed by Deutsche Bank co-chief Anshu Jain who said he was worried the ECB's recent policy measures could remove the incentive for reform. (Editing by Nigel Stephenson)