European shares fell again Tuesday morning, hitting a fresh two-year low for the second straight session as investors worried policymakers had no plan to stem the euro zone debt crisis, which could see Greece default.

Banks gave up early gains that had pushed major indexes higher in early trade.

The STOXX Europe 600 Banking Index fell 1.5 percent, and is down more than 39 percent in 2011. Several heavyweights in the sector are exposed to euro zone debt. BNP Paribas (BNPP.PA) fell 8.1 percent.

People are coming back and focusing on Greece and Italy. It was a relief rally, but it feels like many are looking to sell on any strength. It is very difficult for bulls to hang on there, Joe Rundle, head of trading at ETX Capital, said.

Wall Street turned positive late on Monday, helped by reports after European markets closed that Italy had asked China to make significant purchases of Italian debt, which might help convince markets the euro zone's third-biggest economy can manage its debt load.

Italian economy minister Giulio Tremonti met Chinese officials last week, a Treasury spokesman said on Tuesday.

At 0831 GMT (4:31 a.m. ET), the FTSEurofirst 300 <.FTEU3> index of top European shares was down 0.9 percent at 882.47 points, after a new two-year low of 878.09. It had gone as high as 903.21.

On Monday, the index fell 2.7 percent on intensified worries about the euro zone, including the prospect of Greek defaulting, and with Italian bond yields rising.

The index has lost more than 21 percent in 2011 as investors cut exposure to risky assets such as stocks following an escalation of the euro zone debt crisis, and weak data from major economies that have sparked concern they may go into recession.

An Italian auction of long-term debt due on Tuesday could show if investors have found any reassurance from reports China might offer financial support. Some reports say China will not make invest further in the euro zone.

Traders and strategists were skeptical China's potential bond buying would make a significant difference.

It would not be the first time the market had hoped that China would ride to the rescue, Jeremy Batstone-Carr, strategist at Charles Stanley, said. But the Chinese do not have a great track record. They participated in the Portugal bonds this year, and they lost money.

Across Europe, Britain's FTSE 100 finance/markets/index?symbol=gb%21ftse>.FTSE, Germany's DAX .GDAXI and France's CAC40 .FCHI fell 0.8-2.0 percent.

Italy's benchmark .FTMIB fell 1.1 percent, as bond yields rose.

While the decline in the stock market has made shares look cheap in terms of price-earnings ratios, strategists said the wider picture was too gloomy for investors to have any confidence.

Valuations look attractive, but then we all know that valuations are trumped by the bigger picture when concerns are as elevated as they are, Batstone-Carr said.