The Guardian newspaper reported that France and Germany have agreed to expand a rescue fund. European officials are expected to take up the expansion along with a package of other measures at a meeting this weekend.
The Dow Jones industrial average rose 180.05 points, or 1.6 percent, to close at 11,577.05. It was another day of wild swings for the stock market. The Dow dropped as many as 100 points in the morning and soared as many as 255 points within an hour of the closing bell.
"The news out of Europe is taking fears of a 2008 scenario off the table," said Jeffrey Kleintop, chief market strategist at LPL Financial. The worry hanging over markets for months is that a default by a deeply indebted European government could set off a financial crisis similar to the one triggered by the collapse of Lehman Brothers in 2008.
The S&P 500 index rose 24.52 points, or 2 percent, to 1,225.38. The Nasdaq composite rose 42.51 points, or 1.6 percent, to 2,657.43.
The rally came in stark contrast to the previous day's trading. Stocks slumped Monday after the German government played down hopes that Europe's debt crisis would be resolved soon. It was the worst day for the major indexes since Oct. 3, when all three hit their lowest points in 2011.
Banks and homebuilders also pulled the stock market higher Tuesday. Bank of America Corp. jumped 10.1 percent after it beat earnings expectations for the third quarter thanks to accounting gains and the sale of a stake in a Chinese bank.
Goldman Sachs rose 5.5 percent, even after reporting just its second quarterly loss since going public in 1999.
There was also better news from the housing market, which has rattled banks since the real estate collapse.
A survey of U.S. homebuilders showed they are less pessimistic about the struggling market. The National Association of Home Builders said its index of builder sentiment rose from 14 to 18 this month, the highest level since May 2010. But any reading below 50 reflects overall pessimism.
Markets wavered in early morning trading after some disappointing corporate earnings reports and reports that France and Germany might not reach an agreement on additional support for Greece. An agreement between the two countries is seen as the bedrock for a rescue package that can pass all 17 countries that share the euro.
The ratings agency Moody's also said late Monday that the stable outlook for France's top-notch credit rating is under pressure. On Tuesday, that country's finance minister said that the economy will likely grow a rate of less than 1.5 percent next year. France is Europe's second-largest economy behind Germany.
The Greek government is widely expected to go through some kind of default or restructuring of its debt. If that process becomes disorderly, European banks could suffer big losses on Greek government bonds and that could spread overseas, jolting global credit markets.
Tuesday brought another full day of corporate earnings reports in the U.S. International Business Machines tugged on the Dow average, falling 4.1 percent, the most of any Dow stock by far. IBM reported quarterly revenue that fell short of Wall Street estimates.
UnitedHealth Group Inc. fell 2.7 percent after its third-quarter profit dipped. The country's largest health insurer by sales said medical costs climbed and more patients visited their doctors' office.
Coca-Cola Co. lost half of 1 percent after narrowly beating Wall Street's earnings estimates. Johnson & Johnson rose 1 percent after posting a 6 percent decline in third-quarter profit, roughly in line with analyst expectations.
More than five stocks rose for every one that fell on the New York Stock Exchange. Trading volume was higher than average at 5 billion shares.