At close, London's benchmark FTSE 100 index of top companies gained 1.86 percent to 5,403.28 points, while Frankfurt's DAX 30 rose 1.65 percent to 6,435.60 points and Paris' CAC 40 jumped 1.88 percent to 3,084.09 points. Milan closed 3.4 percent higher and Madrid won 2.10 percent.
In foreign exchange deals, the euro fell to $1.2744 from $1.2815 late in New York on Monday. The dollar rose to 79.98 Japanese yen from 79.30 yen.
US stocks also rose buoyed by news that existing home sales increased in April signaling recovery was under way in the distressed US housing market.
At around 1600 GMT, the Dow Jones Industrial Average was 0.43 percent, the broad S&P 500 index rose 0.73 percent, while the tech-heavy Nasdaq added 0.47 percent.
"No significant negative news from Europe and the latest round of rhetoric on how important it is to keep Greece in the eurozone sent investors on a shopping spree, which helped to lift stock prices for the second day," said trader Anita Paluch at Gekko Global Markets.
"An EU summit on Wednesday, where the idea of Eurobonds is supposed to be put on the table despite Germany's opposition, raises hopes for new debt crisis combating actions, which would include growth promotion."
Attention is now on an informal meeting in Brussels on Wednesday where the crippling debt crisis that threatens the eurozone project will be top of the agenda.
Ahead of the talks, IMF chief Christine Lagarde said the eurozone had made a "serious improvement" in dealing with its sovereign debt crisis but called for member states to do more to support growth.
"More needs to be done in relation to supporting growth, particularly by way of structural reforms, certainly not by way of suggested stimulus," Lagarde said at a press conference in London on Tuesday.
The International Monetary Fund does not believe eurozone countries are in a position to use stimulus to promote growth because "we do not think that the fiscal position of the member states can actually bear that on an aggregate basis," she said.
Recent improvements include structural reforms in Spain and Italy, and moves to increase the eurozone's financial firewall, Lagarde added.
"The market is under the impression that a consensus to do something for growth in Europe is emerging, but we must remain prudent because opinions differ sharply on how we get there," Global Equities broker Yves Marcais said.
Eurozone titan Germany remains fundamentally opposed to the introduction of eurobonds and this will not change despite growing pressure on Berlin to alter its stance, a government source said Tuesday.
"This is not a new discussion ... we think it is the wrong way," the source told reporters on the eve of the EU summit. "This is a fundamental position and it also won't change."
Greece has returned as the key issue in Europe after polls on May 6 saw 70 percent of the electorate vote against pro-austerity parties but with no overall winner.
Many now fear a new vote on June 17 will see a victory for parties who campaigned against a bail-out plan, which would in turn lead Athens to default on its debt obligations and leave the euro.
Asian markets rose Tuesday, following a strong performance on Wall Street and on hopes EU leaders will come to an agreement on dealing with the eurozone debt crisis at an upcoming summit.
Tokyo gained 1.10 percent but shortly after the closing bell, Fitch cut Japan's credit rating by two notches, citing the country's massive public debt as Tokyo struggles to set the world's third-largest economy on a growth track.
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