The euro bounced off a two-month low against the dollar Friday, but the risk of a break below its July trough is seen rising amid a deepening debt crisis.

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 business, 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, Minori Uchida, senior analyst at the Bank of Tokyo-Mitsubishi UFJ, told Reuters.

For now though, sizable buying in the euro against the yen, thought to be from Japanese investors, lifted the euro 0.4 percent against the dollar and the yen in Asia.

The euro rose to $1.3933, after dropping to $1.3873 on Thursday, its lowest in two months. The common currency also gained to 108.00 yen, about a half-yen above its six-month low of 107.54 yen hit on Thursday.

Traders expect the currency to head toward 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 Thursday, saying inflation risks have faded and 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.

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.

Greece had threatened to cancel the deal unless it got 90 percent participation, a stance some banks think may just be a tactic 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 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 slipped to 76.09, 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 0.3 percent to $1.0620, 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 $1.6057, a 61.8 percent retracement of its decline earlier this month.

(Reporting by Hideyuki Sano; Editing by Joseph Radford)