Hopes that the Greek Parliament will support a crucial austerity package and that French banks will roll over their share of Greece's debt supported European stocks on Monday.
Investors will center their attention on Athens, where lawmakers debate another round of tough austerity measures. The Parliament has to pass the euro28 billion ($40 billion) worth of spending cuts and tax rises for the country to get its next batch of bailout funds from last year's euro110 billion bailout package.
If lawmakers fail to back the package in the vote expected on Wednesday, Greece will face a default on its debts.
"The fate of Greece and the evolution of the euro area crisis rest firmly in the hands of the Greek parliament," said Silvio Peruzzo, an analyst at Royal Bank of Scotland.
The markets are hopeful that Greece will get another financial lifeline to see it through the next couple of years even if some lawmakers from the governing Socialist Party fail to back the measures. The expectation is that votes from other parties will see the government home.
That has eased concerns over what impact a Greek debt default would have on Europe's financial system. Many analysts think a default could trigger mass panic in the markets, akin to what happened in the aftermath of the collapse of U.S. investment bank Lehman Brothers back in 2008.
Ahead of the vote, the French government said banks had accepted to roll over a significant amount of their holdings in Greek debt.
French President Nicolas Sarkozy said the plan being worked out between French government officials and bankers would involve reinvesting debt held by French banks in new securities over 30 years. The hope is that it will ease the pressure on Greece to constantly find money to pay off investors.
French bondholders hold about euro15 billion ($21.3 billion) in Greek government debt.
European leaders are trying to get the private sector to take part in continent-wide efforts to help Greece avoid default. Finance ministers from the 17 eurozone countries are scheduled to meet Sunday and confirm Greece's next batch of bailout funds provided the Greek Parliament has backed the austerity measures.
Hopes for a second bailout for Greece have helped stocks in Europe start the week on a firm footing and pushed the euro 0.3 percent higher to $1.4224.
The FTSE 100 index of leading British shares was up 0.4 percent at 5,721 while Germany's DAX rose 0.1 percent to 7,127. The CAC-40 in France was 0.1 percent higher at 3,789.
Wall Street was poised to open positively, too _ Dow futures were 0.3 percent higher at 11,916 while the broader Standard & Poor's 500 futures rose 0.4 percent at 1,269.
As well as keeping an eye on developments over Greece's debt crisis, investors will be monitoring a raft of economic figures this week.
The main release in Europe will be Thursday's first estimate of eurozone inflation in June. Anything higher than May's 2.7 percent rate will likely reinforce speculation that the European Central Bank will raise interest rates even higher after an anticipated quarter point hike to 1.5 percent next week.
In the U.S., the data week culminates in Friday's closely watched Institute for Supply Management survey into the manufacturing sector.
In addition to Greece's debt woes, market sentiment has been undermined recently by concerns that the U.S. economic recovery is slowing down.
Earlier in Asia, Japan's Nikkei 225 fell 1 percent to close at 9,578.31, while South Korea's Kospi lost 1 percent to 2,070.29.
Hong Kong's Hang Seng fell 0.6 percent to 22,041.77, but shares in mainland rose. China's Shanghai Composite Index gained 0.4 percent to 2,758.23 while the smaller Shenzhen Composite Index added 1.1 percent to 1,148.63.
In the oil markets, prices continued to fall following last week's surprise decision by oil-consuming countries to release 60 million barrels of crude over 30 days. Benchmark oil for August delivery was down 75 cents to $90.43 a barrel in electronic trading on the New York Mercantile Exchange.
Pamela Sampson in Bangkok contributed to this report.