* Euro rises on Japanese buying vs yen, outlook shaky
* Break below July low of $1.3838 seen as major bearish sign
* Obama's jobs proposal does little to ease recession worries
* Uncertainty on Greek debt rollover plan could haunt euro
By Hideyuki Sano
TOKYO, Sept 9 (Reuters) - The euro was off a two-month low against the dollar on Friday but the risk of a break below its July trough is seen rising after a deepening debt crisis forced the European Central Bank to drop its tightening policy bias, a key driver in the euro's rally earlier this year.
The market showed a mostly muted response to U.S. President Barack Obama's $447 billion package on jobs that is made up largely of tax cuts for workers and businesses, amid doubts over whether he can push it through a divided Congress.
"The euro now doesn't have the support of expectations for rising interest rates, which clearly points to the higher possibility that the euro will fall below (its July low near) $1.38. In addition, strains on European banks' funding are rising. Given all this, the euro looks likely to fall further," said Minori Uchida, senior analyst at the Bank of Tokyo-Mitsubishi UFJ.
In Asian trade, sizable buying in the euro against the yen, thought to be from Japanese investors, lifted the euro slightly against the dollar and the yen.
The euro traded at $1.3890 , after dropping to $1.3873 on Thursday, its lowest in two months. A close below $1.3900, a 50 percent retracement of the currency's rally from January to May, could raise the case for more weakness.
Traders expect the currency to head towards the July low of $1.38376, a break of which could send a strongly bearish signal, with $1.35 cited as its next possible target.
"The euro is unequivocally bearish. It broke through its long-term support and is likely to go significantly lower," said David Scutt, a trader at Arab Bank Australia.
The European Central Bank held rates steady at its policy meeting on Thursday, saying inflation risks are no longer skewed to the upside and that economic growth in the region will be slow at best, prompting money markets to fully price in a rate cut by the year-end.
Dollar funding strains for European banks showed no sign of abating with the euro/dollar basis swap spread on Thursday hitting its highest since last 2008.
The common currency stood at 107.65 yen , near its six-month low of 107.54 yen hit on Thursday. Against the Swiss franc, it eased to slightly to 1.2130 franc , still above the 1.20 floor set by the Swiss National Bank.
POTENTIAL PITFALL
Another potential pitfall for the euro is uncertainty over Greece's debt swap plan as Friday is the deadline Athens has given investors in Greek bonds to say whether they intend to take part in its debt exchange offer, a key part of a second 109 billion euro bailout package it clinched on July 21 to avoid bankruptcy.
Because of the time needed to process all responses, however, Athens has no plan to announce the result on Friday.
Greece had threatened to cancel the deal unless it got 90 percent participation, a stance some banks think may just be tactics by Athens to get most bondholders on board.
Still, a low participation rate in Greece's debt swap could mean reluctant euro zone partners will have to cough up more cash for the overall package to work.
But the dollar also lacked traction after Obama's long-awaited job proposals failed to boost hopes of a U.S. recovery. U.S. jobless claims unexpectedly rose last week, highlighting the fragile state of the U.S. job market.
"To some extent, this was largely in line with the chatter we heard before it's release. It may even be a bit smaller than needed given the gravity of the problem. That could prevent markets from reacting too positively," said Omer Esiner, senior market analyst at Commonwealth Foreign Exchange in Washington.
"And at the end of the day, it depends on what the finished product will be. A lot of this will be chopped up before it is passed. We've seen a lot of political paralysis in Washington."
Federal Reserve Chairman Ben Bernanke offered little new insight as to what the central bank will do at its policy meeting on Sept. 20-21 in his speech on Thursday, though most players remain convinced that the bank will start buying longer-dated bonds in a bid to try to lower longer bond yields.
The dollar index stood at 76.23, having surged to two-month highs of 76.319 on Thursday. Against the yen, the dollar stood flat at 77.48 yen .
The Australian dollar gained 0.2 percent to $1.0600 , but lacked the energy to tackle a resistance-packed zone from $1.0630, its 55-day moving average, through $1.0648, the 100-day average, to $1.0657, a 61.8 percent retracement of its decline earlier this month. (Additional reporting by Cecile Lefort in Sydney; Editing by Joseph Radford and Chris Gallagher)