Stocks rose on Tuesday, extending a global rally into a third consecutive day, as investors anticipated a more ambitious plan by European leaders to deal with the euro zone debt crisis. But they were unable to hold on to their sharpest gains.

After jumping 2.5 percent on Monday, the Dow Jones industrial average added 1.33 percent Tuesday, rising 146.83 points, to 11,190.69. Earlier in the day, the Dow was up as much as 325 points. But the markets lost steam in late trading on concerns that a second bailout for Greece could be scuttled by countries that wanted banks to shoulder more of the burden.

The euphoria at the start of the week centered on talk of a plan to give extra firepower to Europe’s proposed 440 billion euro (about $600 million) bailout fund for troubled nations and banks. Under a possible new plan, the fund could be enhanced to insure as much as a couple of trillion euros in loans.

Even if leaders manage to eventually expand the war chest, it is far from certain that the larger crisis in Europe will be contained. For one thing, the money might not be enough if the already weakened European economy falls into a prolonged recession. Recent data show that manufacturing and services activity there are sliding toward contraction, and that the region’s economy probably stagnated in the third quarter.

As it is, Greece, Portugal and Spain are looking at lengthy economic downturns as a result of harsh austerity measures adopted to straighten their finances. In Italy, economists are increasingly worried that that a 45 billion euro ($61 billion) austerity package recently rushed through Parliament could tip the economy into a downturn.

Germany and France also have started to show signs of a slowdown, a more worrisome development since they are considered engines of the European economy.

“They are buying time,” said Jens Nordvig, an economist at Nomura in New York. “But it only postpones the problem. Growth is slowing in countries like Italy. They may be too late.”

The immediate hurdle for each of the 17 countries in the euro currency zone is to approve the 440 billion euro fund proposed in July. So far only seven have done so. Slovenia signed off on Tuesday and there are critical votes set for Wednesday in Finland and in Germany on Thursday.

European leaders say the process should be completed by mid-October but already investors judge it insufficient to cope with the scale of problems in Greece and other troubled nations.

As a result, a new plan is emerging in Europe that would let the European Central Bank or another European institution, like the European Investment Bank, buy troubled sovereign debt or loans from euro zone banks with the backing of the fund.

Timothy F. Geithner, the Treasury secretary, turned up the heat on Europe in the last few days to expand the war chest exponentially by leveraging its resources, possibly by running it like a bank that could borrow additional money.

The governor of Canada’s central bank, Mark Carney, said last week that Europe should make about one trillion euros available to “overwhelm” the crisis.

Lorenzo Bini Smaghi, a member of the executive board of the European Central Bank, offered cautious backing to the idea on Tuesday, saying leaders “need to look at possible leverage.” The bank’s support would probably be essential to such a plan.

But critics said that using it to buy distressed government debt held by banks would only shift the problem to Europe’s taxpayers. That would worsen the region’s sovereign debt crisis and potentially compromise even relatively strong nations like Germany and France.

Other Europeans are grumbling that the United States is hardly in a position to lecture them on how to manage a crisis, and it is not clear how quickly or easily the fund could be expanded. Germany, for one, does not want the independent central bank to be involved in the fund’s operations.

Still, no politician wants to pump even more taxpayer money directly into the fund, which would anger voters and put the AAA ratings of Germany and France at risk. Wolfgang Schaüble, the German finance minister, on Tuesday derided the idea as “silly.”

The prospect of a new plan, which emerged during a weekend of meetings of world leaders in Washington and after some urging by the United States, seemed to represent a realization that Europe had to do something to bring itself back from the brink. For weeks, global markets had lurched downward and investors had become increasingly concerned that Europe’s failure to act was disrupting confidence and growth around the world.

On Tuesday in Europe, the Euro Stoxx 50 index, a barometer of euro zone blue chips, closed up 5.3 percent.

Later, the United States markets gave up some of their gains after The Financial Times reported that some countries were demanding that banks take a bigger hit in Greece’s debt restructuring and that Greece needed deeper financing than was thought two months ago. The Standard Poor’s 500-stock index rose 1.07 percent, or 12.43 points, to 1,175.38, and the Nasdaq composite index gained 1.2 percent, to 2,546.83.

Earlier Tuesday, Mohamed A. El-Erian, chief executive of Pimco, who has long been pessimistic about Europe’s efforts to solve its crisis, expressed optimism for a solution. “They recognize they have deep problems and they recognize they need to do something about it,” he said in a Bloomberg radio interview. “This was a very important wake-up call for Europe.” He added that European leaders “finally get it.”

But skepticism remained.

“It is not clear to me that, though they may have ‘got it,’ anything has actually happened,” said Carl Weinberg, an economist at High Frequency Economics. “At this point, you have got to show me the money.”

Joshua Brustein contributed reporting.

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