* Ukraine has started to pay off gas debts
* Gazprom delayed switching Kiev to prepayment
* Tension eases after talks, more planned (Adds talks continued on Tuesday, Gazprom says exports to Europe up)
By Vladimir Soldatkin and Denis Pinchuk
MOSCOW, June 3 (Reuters) - Russian and Ukrainian energy companies tried again on Tuesday to settle a dispute over unpaid gas bills that is threatening supplies to Europe and stoking a political conflict setting Moscow against Kiev and the West.
Russian state-owned exporter Gazprom gave Kiev some respite on Monday by allowing it six more days to pay its debts, averting the risk of an immediate cut in supplies under an advance payment system that had been due to come into force.
Although the two sides disagree over the price Ukraine should pay, signs of compromise are growing after the latest talks brokered by the European Union in Brussels on Monday.
"Talks went for a long time yesterday ... There is an understanding on how to move forward - this is very important," Gazprom spokesman Sergei Kupriyanov told reporters in Moscow.
Talks between Gazprom and Ukrainian energy company Naftogaz continued on Tuesday in Berlin, a source at Gazprom said.
Ukraine owes billions of dollars and Prime Minister Arseny Yatseniuk hinted that his cash-strapped country would be now be more flexible over its price demands, although he made clear the price sought by Russia remained far too high.
"We look forward to the completion of negotiations with Russia's Gazprom this week," Yatseniuk told parliament in Kiev. "We understand it will be hard to reach $268 but we will never accept $500."
Ukraine wants to return to a discounted gas price of $268.5 per 1,000 cubic metres, granted by Moscow as a concession after Ukraine's then president Viktor Yanukovich dropped plans to sign agreements on closer political and trade ties with the EU.
After Yanukovich was ousted in February following months of protests provoked by his policy U-turn, Russia reinstated the earlier price of $485 - the highest in Europe.
Russian President Vladimir Putin does not want tensions over the gas dispute to overshadow a visit to France this week where he is likely to come face to face with U.S. President Barack Obama and is planning to meet European leaders who have imposed sanctions on Russia over its annexation of Crimea from Ukraine.
Ukraine's newly elected president, Petro Poroshenko, is attending the same event marking the 70th anniversary of the D-Day landings that opened the Western front against Nazi Germany in World War Two. His inauguration is set for Saturday.
Despite the signs of compromise, Yatseniuk reiterated the threat of a legal challenge in the Stockholm arbitration court if no agreement is reached.
European Energy Commission Guenther Oettinger is brokering attempts to compromise on the price, with the average gas price paid by Europe to Gazprom at around $370 in 2013.
Late on Monday, after some six hours of talks, Oettinger said the chief executives of Gazprom and Ukraine's Naftogaz had agreed to consider a plan that could avoid price disputes recurring over the European winter when demand peaks.
Gazprom accounts for around a third of Europe's gas needs and supplies to Europe carried via pipelines across Ukraine were disrupted during two previous "gas wars".
Gazprom CEO Alexei Miller sought to emphasise the importance of Russian gas to Europe, saying supplies to Europe and Turkey in May had risen by more than 10 percent in May and by more than 5 percent in the first five months of 2014, year-on-year.
"At the same time, as we predicted, domestic gas production in Europe and supplies from other sources continue to decrease," Gazprom quoted Miller as saying.
Separately, however, Gazprom said it planned to export 158.4 billion cubic metres (bcm) of gas this year to Europe and Turkey, down from 161.5 bcm in 2013. But a senior official said this was a "conservative" forecast which could be revised later.
Russia said on Monday that Kiev, facing the prospect of its gas debt to Moscow rising above $5 billion by June 7, had paid $786 million of its bill.
The moves helped reduce tensions in the gas row, despite continuing political recriminations over fighting between pro-Russian separatists and the Ukrainian armed forces that are trying to regain control of territory in east Ukraine.
"Our view is that Gazprom and Naftogaz will reach a compromise on the price at around $350; however, it is unclear how long the discussions will continue and what the possible consequences will be," Russia's VTB Capital said in a research note. (Additional reporting by Pavel Polityuk in Kiev and Steve Gutterman in Moscow, Writing by Katya Golubkova, Editing by Timothy Heritage/Ruth Pitchford)