US shares staged a late recovery on Monday to post only their second positive close of the month.
Earlier, European and Asian markets fell on fears that Greece may default.
A series of news reports that Germany may be preparing for an "orderly default" by Greece also sent the euro lower.
German officials sought to shore up confidence on Monday, saying the stability of Greece and the euro was "the common goal".
Bank shares were hardest hit, with France's BNP Paribas closing down 12%.
Other French banks also fell amid rumours of a possible downgrade of their debt and concerns about their exposure to Greece and Italy.
London's FTSE 100 closed down 1.6%, France's Cac 40 dropped 4.0% and Germany's Dax lost 2.3%. The declines followed heavy falls in Asia, where Hong Kong ended 4% down.
But having spent most of the day in negative territory, the Dow Jones Industrial Average closed up 69.0 points, or 0.63%, at 11,061.1.
The euro fell to a 10-year low against the yen, and investors poured money into German bonds in a flight to safety.
The latest crisis of confidence in the markets came amid worries that Germany had lost patience with Greece - and other heavy indebted eurozone nations - and might not help future bailouts.
Germany's Economy Minister Philipp Roesler said in a newspaper article at the weekend that an "orderly default" by Greece could no longer be ruled out.
On Monday, adding to the tensions, the general secretary of German Chancellor Angela Merkel's junior coalition partner suggested that Greece could leave the eurozone.
"In the final analysis, one also cannot rule out that Greece either must, or would want to, leave the eurozone," Christian Lindner, the general secretary of the Free Democrats (FDP), said in a television interview.
This followed Friday's surprise resignation of the European Central Bank's (ECB) chief economist, Juergen Stark.
His departure was seen as a sign of divisions within the ECB and among eurozone leaders over what to do about Europe's debt crisis.
On Monday, a spokesman for Mr Roesler, who is also vice-chancellor, tried to dampen the impact of his newspaper comments.
"Our common goal is the stability of the euro and we want Greece to stay in the euro," the spokesman said.
At the same news conference, Mrs Merkel's spokesman said that Germany "assumes that Greece is doing everything it can" to implement strict austerity measures to battle its deficit woes.
"Our goal is quite clear: we want to stabilise the eurozone as a whole," he said.
But stock markets remained deep in the red, especially French banks' shares, which are among the most exposed to Greek debt.
France's two other big banks, Societe Generale and Credit Agricole, closed down 11%. In Germany, Deutsche Bank fell 7.3% and Commerzbank 8.3%.
UK banks escaped relatively lightly, helped by the release of the Vickers report on breaking up UK banks and a belief among some investors that the recommendations may be watered down.
HSBC closed down 2.4%, Lloyds dropped 1.6%, RBS fell 3.4%, and Barclays was 1.6% lower.
The euro fell to 104.09 yen, its lowest since June 2001. The euro was also down against the dollar.
Germany's cost of borrowing for 10-year bonds fell to historic lows on Monday, as investment funds fled riskier assets.
The yield - or interest rate - indicated by the price of German 10-year bonds fell to 1.709% from 1.770%.
Satoshi Tate, a currency dealer at Mizuho Corporate Bank, said: "We are watching Greece and only Greece.
"Conditions are getting very serious and everyone is worried how the issue will unfold," he added.
Marc Ostwald, market strategist at Monument Securities, added: "With German officials seemingly in destructive overdrive, as per all the public talk of preparing for a Greek default and even a Greek euro exit, markets can hardly be blamed for the latest charge for the bunker and tin hats."