Wary investors looked for bargains after recent sharp losses on Friday, keeping shares and the euro just above multi-month lows.
They were also balancing hopes of policy action to shore up Europe's ailing economies against concerns about collateral damage if Greece leaves the single currency.
U.S. stocks were tipped to open up.
German consumer morale held steady going into June while Chinese exports showed signs of recovery in early May, countering dire recent data that suggested Europe's growth engine was no longer immune from the region's debt crisis and factory output in the world's number two economy was faltering.
The modest data boost helped the euro rise 0.3 percent to $1.2583, inching up from two-year lows of 1.25155 as bearish investors took a breather from a sharp sell-off. But the common currency stayed on track for its fourth straight week of losses.
Safe-haven flows drove the index that measures the dollar against key currencies to a fresh 20-month peak of 82.411 during early European trade.
"Markets have priced in a very negative scenario for Greece as well as deteriorating growth prospects in the euro zone, but with them very much focused on the tail risk of Greece leaving the euro bloc, the euro remains highly vulnerable," said Masafumi Yamamoto, chief FX strategist at Barclays.
The FTSEurofirst 300 index inched up 0.1 percent to 983.50 points, adding to a 1.1 percent gain on Thursday that took it further away from its May 21 trough of 952.55 points, its lowest point since December 20.
With the euro zone mired in crisis, speculation that European policymakers might soon intervene again boosted some shares.
Bank stocks, which have fallen sharply on their exposure to the debt turmoil and fears of contagion from any Greek euro exit, were among the best performers, with the euro zone banks index up 0.65 percent.
Credit Agricole said the European Central Bank (ECB) could announce new stimulus measures next month, such as another round of emergency funding for banks in the region.
"We expect the ECB to make a move (at its next rate-setting meeting) on June 6," it said in a research note.
ECB action then could help lift the prevailing mood of uncertainty among global investors who remain sharply focused on June 17 elections in Greece that will largely determine if Greece gives up the euro.
One opinion poll published on Thursday showed the anti-bailout leftist party SYRIZA maintaining its lead ahead of the vote.
The unclear outcome of the ballot has also driven debt markets, sending yields on safe-haven German bonds to record lows and culminating on Wednesday in a sale of two-year Bunds that drew strong demand despite carrying an unprecedented zero coupon.
On Friday, a tentative search for higher returns prompted a sharp rally in non-German debt, with the yield on French 10-year bonds - viewed as a viable alternative to Bunds - falling to 2.43 percent, matching its euro-era low.
Other euro zone sovereign yields also fell, while the equivalent benchmark German yield was 1.40 percent, or 1 basis point higher - though German debt was not expected to lost its appeal.
"This is all built on very shaky ground, and we could easily see things reversing again," said DZ Bank strategist Michael Leister. "The (debt) market is trading political headlines which means, in turn, the market will remain very volatile for the time being."
Investor jitters also drove oil markets, with Brent crude inching up 50 cents to $107.05 a barrel but staying on track for a fourth weekly loss and its longest losing streak since early 2010 as concerns about slowing global economic growth blunting demand dominated.
With trading subdued and U.S. markets closed on Monday for Memorial Day, some said the rally in assets might peter out before the end of the day.
"Given the uncertainties in the financial markets related to Greece, there may be an interest in cutting risk exposure ahead of the weekend," Derek Halpenny, European Head of Global Markets Research at Bank of Tokyo-Mitsubishi UFJ, said in a note to clients.
The underlying risk-off mood earlier saw Asian shares outside Japan post a third consecutive week of losses to hit their lowest levels of the year.