* Euro hits session low of $1.4055 after ECB's 1-year tender
* Euro up 0.2 percent at $1.4104
* Fed seen dampening rate hike prospects, questions over QE
* Yen falls as cross rebound extends
(Adds quotes, updates prices)
By Tamawa Desai
LONDON, June 24 (Reuters) - The euro briefly fell on firm demand at the ECB's first-ever offer of cheap one-year funds on Wednesday, while the dollar was broadly lower on expectations the U.S. Federal Reserve will emphasise it is in no hurry to raise rates.
The European Central Bank said it will lend banks 442.24 billion euros ($612.8 billion) in its first one-year refinancing operation, marking its biggest ever liquidity injection.
The allotment is above the expectations of traders in a Reuters poll, who forecast the ECB would allot 300 billion euros, and exceeds the previous record allotment of 348.6 billion in December 2007.
"The large drawdown is strongly positive for market liquidity as it exceeds the minimum required reserves that banks need to raise during the current maintenance period by some 250 billion euros according to our estimates," said Lena Komileva, head of G7 market economics at Tullett Prebon.
"This will put immediate downward pressure on short-end Eonia rates and drag 3-month Euribor down, which is bad for the euro," she added.
A European money market trader noted talk that much of the money obtained in the ECB operation would go into banks' dollar books and be sold or swapped for dollars.
The euro briefly hit a session low of $1.4055 immediately after the results. But by 0943 GMT, it was back up, rising 0.2 percent on the day at $1.4104. Earlier the euro hit a session high of $1.4138, its highest since June 11, according to Reuters data.
In a positive technical signal, the euro on Monday broke its downtrend line from early June, after holding on a closing basis support at 1.3782, the 38.2 percent retracement of its rise from April. This indicated many people still expect medium-term appreciation of the euro and other currencies against the dollar to fresh highs in this cycle.
The euro was also helped by comments from ECB officials hinting euro zone interest rates would not fall further.
WATCHING THE FED
The Fed began its two-day meeting on Tuesday, and many expect policymakers will want to play down expectations of higher interest rates that have built in the market, for fear these will choke an economic recovery.
That pushed the dollar broadly lower, with the dollar index at its lowest since June 12. It was last down 0.3 percent at 79.60.
Currencies such as the Australian dollar, which have been favoured as recovery plays against the dollar and yen, leapt 1 percent or more on Tuesday as the greenback lost favour.
The Aussie dollar was last up 0.8 percent to $0.8008.
The Fed, which makes its announcement at 1815 GMT, will also be under scrutiny for what it says about its asset buying programme aimed at keeping longer-term interest rates down.
The market will also keep an eye on the week's record $104 billion auction of U.S. Treasury notes, which kicked off with exceptionally strong demand for the $40 billion issue of two-year paper, including a record indirect bid.
But there was some anxiety for the five- and seven-year Treasury notes to be auctioned on Wednesday and Thursday.
The dollar fell slightly against the yen at 95.10 yen.
The Japanese currency was down against a range of currencies due to talk about several big investment trusts expected to be launched on Wednesday which were likely to attract plenty of funds for investment abroad.
"Capital outflows from Japan will likely keep pressure on the yen," said Lee Hardman, currency economist at Bank of Tokyo-Mitsubishi UFJ.
The OECD said the economic outlook has improved for the first time in two years but soaring unemployment and ballooning budget deficits could knock a weak recovery off track.
The Organisation for Economic Cooperation and Development said in its latest economic outlook that the slowdown in its 30 member countries was close to the bottom. Anaemic growth of 0.7 percent should return to the area next year after a 4.1 percent fall this year. (Additional reporting by Jessica Mortimer; editing by Stephen Nisbet)