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Although Wall Street saw record falls on Monday after Congress blocked the deal, the main Dow Jones index rose 271 points or 2.6 percent in early trading.
Analysts said investors are hopeful a new deal can be agreed this week.
Bush warned that if agreement is not reached, the U.S. economy faces "painful and lasting damage".
While the Dow Jones was up 2.6 percent to 10,637 in morning trading, the other main Wall Street index - the Nasdaq - had advanced 2.8 percent.
Despite their strong gains, European shares remained mixed in afternoon trading.
While the UK's FTSE 100 was up 23 points or 0.5 percent to 4,842, Germany's Dax was down 60 points or 1 percent to 5,748.
Meanwhile, Japan's Nikkei index ended Tuesday down 4.1 percent, while Hong Kong's Hang Seng rose 0.8 percent.
U.S. stocks jumped on Tuesday, a day after Wall Street's worst day in 20 years, as investors voiced optimism Congress will eventually pass a plan to rescue tainted banking assets, lifting the U.S. dollar and crude oil.
U.S. government debt prices slipped, paring Monday's frantic charge for safe-haven investments. Oil rebounded more than $2 a barrel toward $98 after nearly a 10 percent drop the previous session as investors were less fearful of a major meltdown in capital markets.
The dollar jumped as much as 1 percent against the yen as cautious optimism for approval of a $700 billion plan to bail out banks replaced the shock following Monday's rejection by the U.S. Congress.
BLACK MONDAY
The Dow lost 778 points on Monday, falling 6.98 percent, its largest point decline in history, and posted its biggest daily percentage slide since the 1987 stock market crash. The benchmark S&P 500 also had its worst day in 21 years after the House voted down the bailout plan by a count of 228 to 205.Â
World stocks, as measured by MSCI's entire country world index, on Monday lost about $1.7 trillion for the day.
The failure of the bill, which would have let the Treasury buy up bad mortgage debt from struggling banks in an effort to kick-start much needed lending, was seen as crucial to shielding the economy from an even deeper slowdown.
The bailout's demise comes after U.S. bank Wachovia was forced to sell most of its assets to Citigroup in a deal brokered by the Federal Deposit Insurance Corp.
That followed fast upon fresh signs that financial market turmoil was spreading around the world. European authorities in recent days were forced to step in and rescue a group of banks in Britain, Belgium, Germany and elsewhere.
Global money markets remained paralyzed, even as central banks, including the Federal Reserve, pumped cash into world markets in an attempt to boost liquidity.
Although there were doubts that the government's rescue package would be sufficient to shelter the economy and stem the spread of the turmoil, investors said it was a necessary first step to restoring confidence in financial markets.
SHOCK VOTE
The House of Representatives voted 228-to-205 against a compromise bailout plan that would have allowed the Treasury Department to buy up toxic assets from struggling banks. House Republicans, in particular, balked at spending so much taxpayer money just before the Nov. 4 U.S. elections.
"I can't believe they weren't able to come together and come up with a solution. Complete disaster was predicted if it didn't pass," said Stephen Berte, senior equity trader at Standard Life in Boston. "I can't see what the upside is right now," he was quoted by Reuters as saying.
Investors rushed to assets considered a safe haven. Government bond prices and gold jumped, and oil fell below $99 per barrel on the view that world demand will contract as the financial crisis puts the brakes on economic activity.
U.S. BAILOUT PROSPECTS UNCERTAIN
In Washington, the failure of the bailout bill -- after more than a week of intensive closed-door negotiation intended to hammer out a compromise plan -- brought new uncertainty about the response of the U.S. government to the worst financial crisis since the Great Depression.
U.S. President George W. Bush was set to huddle with economic advisers to consider the administration's next move after the White House failed to win support for the bailout plan from Bush's fellow Republicans.
"There's no question the economy is facing a difficult crisis that needs to be addressed," White House spokesman Tony Fratto told reporters.
The bailout plan was announced by the Bush administration a little over a week ago. Republican House members voted against it by a more than 2-to-1 margin. A majority of Democrats voted in favor.
Both parties blamed each other for the failure of the closely watched bill after hours of closed-door negotiations intended to add provisions to protect taxpayers and head off criticism that Washington was riding to the rescue of bankers many Americans blame for triggering the housing crisis.
The high-stakes political showdown on the bailout proposal came after Wachovia Corp agreed to sell most of its assets to Citigroup Inc in a deal brokered by regulators. It was one of three U.S. financial deals struck as the crisis deepened.
GLOBAL CONTAGION
Investors said there were ample signs that a financial crisis that started with risky lending to the overheated U.S. property market had gone rapidly global.
"The crisis is going to affect everybody. It's a very difficult situation and it's going to affect economies everywhere," Mexican billionaire Carlos Slim said.
Earlier, the governments of Belgium, the Netherlands and Luxembourg moved to partly nationalize Belgian-Dutch group Fortis NV, and German lender Hypo Real Estate Holding AG secured a credit line from the German government.
British mortgage lender Bradford & Bingley Plc was brought under the government's wing, shares of French bank Dexia tumbled on a report that it might need emergency capital, and bank rescue deals also emerged in Iceland, Russia and Denmark.
The world's central banks, led by the U.S. Federal Reserve, announced a $330 billion expansion of currency swap arrangements, which allows them to increase the amount of money they can provide in their home markets, effectively throwing more money at the crisis.
The Wachovia deal is the latest in a series of events that has transformed the American financial landscape and wiped out hundreds of billions of dollars of shareholder wealth.
The changes include the government takeover of mortgage finance companies Fannie Mae and Freddie Mac, the bankruptcy of Lehman Brothers Holdings Inc, the failure of giant savings and loan Washington Mutual, and Bank of America Corp's purchase of Merrill Lynch & Co Inc.