* Syriza demands early elections, say on policy decisions
* Markets rise on relief govt not in danger of collapse
* Snap election possible before presidential vote early 2015 (New story with politicians' quotes, government and EU officials)
By Karolina Tagaris and Deepa Babington
ATHENS, May 26 (Reuters) - Greece's ruling coalition will find it harder to push through unpopular reforms demanded under an EU/IMF bailout after a clear EU election victory by the radical leftist Syriza party.
Syriza's win by nearly four points fell short of the margin needed to threaten Prime Minister Antonis Samaras's government, and financial markets rose on relief the pro-bailout coalition will be in place to negotiate debt relief talks later this year.
But an emboldened Syriza has promised to make life even more difficult for a fragile right-left coalition hanging on to a slim two-seat majority in the 300-seat parliament.
In an early sign of the troubles ahead, Syriza leader Alexis Tsipras demanded early elections, declared parliament had lost its legitimacy and said Samaras should get his consent before appointing the next governor of the Bank of Greece in June.
"This parliament has no legitimate right to make crucial decisions which will bind the country and the people for the following years," Tsipras told reporters after meeting Greece's president on Monday to discuss his election win.
"We must go to national elections as soon as possible in a coordinated and normal way for democracy to be restored."
Greece still needs to push through pension and other reforms to be eligible for the final aid payments under its EU/IMF bailout which expires this year and to position itself ahead of debt relief talks.
Syriza also bagged the post of governor of the wider Athens region in local elections, allowing it to have a say in some key privatisations, which it opposes.
"This is a victory for the unemployed citizens, for the pensioners," Syriza's Rena Dourou, elected governor of the greater Athens region, told Greek TV. "One could say this is a victory of David over Goliath".
Samaras has softened his pro-bailout message in recent months while urging Greeks to stick it out as the country returns to economic growth after a six-year recession this year.
Against this backdrop, European lenders may be more lenient with Greece during autumn talks to reduce the country's debt burden, to avoid further endangering Samaras's government, a European official familiar with Greek bailout talks said.
"The election's outcome may soften lenders' stance in the debt relief talks," the official said.
The Athens stock market rose 1.5 percent and Greek 10-year bond yields fell 28 basis points on Monday, heading for its biggest daily fall in a month.
With 99 percent of the vote counted, Syriza had 26.6 percent of the vote to become the first radical leftist party to win at the national level in modern Greece, ahead of Samaras's New Democracy conservatives who took 22.7 percent.
"This is a historic win," Tsipras told flag-waving supporters in the early hours of Monday. "Today, the whole of Europe is talking about Greece because it condemned austerity."
The coalition of New Democracy and PASOK combined won a bigger share of the vote than Syriza, which analysts said averted the possibility of early elections until a presidential election in early 2015 muddies the political landscape again.
Greek law says parliament must be dissolved if no presidential candidate secures 180 parliamentary votes - a level of backing that Samaras does not have.
Analysts say if Samaras is unable to find alliances to elect a new president, then he may opt to call snap elections as early as later this year to cash in on a boost in Greece's economic fortunes.
"The election result doesn't clear the horizon around the presidential election in spring," the European official said. "This creates uncertainty and might hurt Greece-bound investment."
For now, Samaras is expected to reshuffle his cabinet following the election defeat, with the focus on whether Finance Minister Yannis Stournaras keeps his post or heads to the Bank of Greece as governor, government officials said.
"The reshuffle won't be immediate, it won't happen in the coming days or this week," said one official, who declined to be named. (Additional reporting by Harry Papachristou, George Georgiopoulos, Angeliki Koutantou and Renee Maltezou. Writing by Deepa Babington, editing by Mike Peacock)