* MSCI Asia ex-Japan adds 0.5 pct, Nikkei trims gains to rise 0.4 pct
* Euro falls, stays above $1.26 vs dollar
* Firmer Tankan nudges yen up slightly vs dollar
* Weak China PMI pushes crude oil futures down $1
* European shares likely to rise a touch
By Chikako Mogi
TOKYO, July 2 (Reuters) - Asian shares rose on Monday with sentiment brightening at the start of the third quarter after European leaders agreed to shore up the region's troubled banks, but the euro gave up some of its gains amid concerns that the debt crisis is still far from over.
MSCI's broadest index of Asia-Pacific shares outside Japan added 0.5 percent, after soaring 2.7 percent on Friday for its biggest one-day rise in more than six months. The MSCI Asia ex-Japan share index ended April-June down 7.4 percent, after rising for the previous two quarters.
European shares were likely to mostly rise marginally, with spreadbetters predicting that region's major markets would open up as much as 0.3 percent. U.S. stock futures were down 0.2 percent.
Japan's Nikkei average trimmed early gains to stand up 0.4 percent, but kept above the key 9,000 level, which it reclaimed on Friday for the first time in 7 weeks.
Strong short-covering spearheaded Friday's broad-based rally as market players who had not expected European leaders to strike any fresh deals scrambled to cover negative bets, and fund managers remained cautious of the sustainability of "risk-on" momentum.
"Even though markets went up significantly on Friday night, we are still fairly cautious as to whether that's a sustainable move, given that the announcements from the summit were probably better than the actual detail of what's been delivered," said Andrew Pease, Sydney-based chief investment strategist at Russell Investments Asia Pacific.
The euro fell 0.3 percent to $1.2627, after spiking 1.7 percent against the dollar to a high of $1.2693 on Friday, its biggest one-day jump in eight months.
"Their agreement offered a direction but details are yet to be worked out, and that will limit the euro's upside," said Yuji Saito, director of foreign exchange at Credit Agricole Bank in Tokyo. "Still, with concerns over the euro zone's debt crisis easing for now, the focus this week turns to U.S. data, including the monthly jobs report."
The dollar eased 0.1 percent to 79.76, after a closely watched Bank of Japan survey on Monday showed big Japanese manufacturers' sentiment improved in the second quarter from the previous quarter.
Euro zone leaders agreed on Friday to let their rescue fund inject aid directly into stricken banks from next year and intervene on bond markets to support troubled member states.
The step relieved credit market strains that had threatened to force struggling Spain into seeking an international bailout, pushing Spanish and Italian debt yields sharply lower.
European Union leaders also pledged to create a single banking supervisor based around the European Central Bank in a landmark first step towards a European banking union.
But Russell Investments' Pease and others were less than convinced that the agreement would be implemented smoothly, paying particular attention to the size of rescue funds which they see as insufficient to bail out needy member states.
"There's still a long way to go and the sorts of things that we would need to make us confident that Europe is no longer a huge hindrance on markets is a credible plan to provide a liquidity backstop for sovereigns and a big enough fund to convincingly recapitalise banks across the region," Pease said.
The EU's joint statement said that the euro zone's temporary EFSF and permanent ESM rescue funds would be used "in a flexible and efficient manner in order to stabilise markets" to support countries that comply with EU budget policy recommendations.
"In theory if these funds were sufficiently capitalised there would be cause for celebration. Unfortunately, they are not sufficiently capitalised and therefore their is no cause for celebration," said Jeff Sica, president of SICA Wealth Management, which manages more than $1 billion in client assets, real estate and private equity holdings.
"The irony of this ... is that those intended to fund it are the very ones who need it."
U.S. crude fell more than $1 or 1.3 percent to $83.90 a barrel, after shooting up $7.27 on Friday, the fourth largest daily gain in dollar terms since the contracts were launched. Brent shed more than $1, falling 1.5 percent at $96.35 after posting its biggest one-day rise since April 2009 on Friday with a rise of more than $6.
Oil futures were hit by data over the weekend showing China's government purchasing managers' index fell to 50.2 in June to a seven-month low, suggesting growth in manufacturing sector activity at big, state-backed firms was close to stalling. Shrinking new orders and the steepest fall in export orders since December pointed to no immediate pick-up for the world's second-biggest economy.
"Chinese data is one of the contributors to the softer turn this Monday, but I think the oil market has had time to think about the implication of the EU deal over the weekend and is reacting now," said Ric Spooner, chief market analyst at CMC Markets.
A private sector survey showed on Monday that China's factory activity at smaller, private-sector companies also shrank in June at the fastest pace in seven months as new export orders tumbled to depths last seen in March 2009.
Later on Monday, purchasing managers' data is due to be released from Europe and the United States, as well as India.
Asian credit markets firmed, pushing the spread on the iTraxx Asia ex-Japan investment-grade index narrower by 3 basis points.