* Gas central to relations between Russia, Ukraine and EU
* Gazprom has extended payment deadline to June 16
* Ukraine says Russian offer is "a trap" (Updates with end of talks, adds quotes)
By Barbara Lewis and Alexei Anishchuk
BRUSSELS/NOVO-OGARYOVO, Russia, June 11 (Reuters) - Russian President Vladimir Putin accused Ukraine on Wednesday of forcing gas talks into a "dead end" by rejecting the offer of a cut in duty to resolve a price dispute that threatens supplies not just to Ukraine but to the rest of Europe.
Talks ended with Kiev demanding contract changes to bring down the highest prices in Europe for Russian gas supplies, and Moscow suggesting its proposed cut of about one-fifth to around $385 per 1,000 cubic metres was its final offer.
Further negotiations could take place by phone before Monday, when Moscow has threatened to halt supplies.
The dispute is part of a broader stand-off between Ukraine and its former Soviet master, as Kiev's new West-leaning leadership struggles to contain a pro-Russian separatist rebellion in its eastern provinces.
Arguments over how much Russia charges for its gas have rumbled on for years and led to supply cut-offs in 2006 and 2009.
Putin, meeting his government outside Moscow, said Russia had offered a discount of $100 per 1,000 cubic metres by removing an export duty. That would roughly bring it in line with the price paid by other European consumers.
"The reduction is too little for our Ukrainian partners; they want more, although it's not clear on what basis," he said. "If this is the case, then it seems that the issue is being driven into a dead end."
The latest crisis flared with the overthrow of Ukraine's Moscow-leaning president Viktor Yanukovich in February, after which Moscow annexed its neighbour's Crimea region and nearly doubled the price Kiev has to pay for gas.
Russian gas exporter Gazprom had been charging a preferential rate as part of Moscow's effort to keep Ukraine from drifting into the orbit of the European Union.
Earlier on Wednesday, Ukrainian Prime Minister Arseny Yatseniuk made clear at a cabinet meeting that Kiev rejected Russia's offer to lower gas prices by removing the export duty, a measure that would not be written into the contract and was therefore at the whim of Moscow.
"I want to say that we know these Russian traps. The discount is set by the Russian government and cancelled by the Russian government," he said.
Ukraine instead wants to change a 2009 contract that locked it into buying a set volume of gas, whether it needs it or not, at $485 per 1,000 cubic metres. The average gas price for Europe was about $387 per 1,000 cubic metres last year, but Gazprom has given no current figure.
The European Commission, the EU executive, has taken on the role of brokering trilateral price talks, but six rounds have failed to secure a deal.
Arriving for the latest talks on Wednesday at the European Commission in Brussels, EU Russian envoy Vladimir Chizhov hinted that Ukraine was the obstacle.
"These negotiations stand a good chance of success, but this particular tango takes three," he said. Asked who was out of step, he said: "No prizes for guessing."
Although Wednesday's more than four hours of negotiations stopped short of a deal, the European Commission said the parties would stay in touch by telephone to try make progress over the next 48 hours.
That timeframe gives scope for Ukraine to get any payment to Moscow by the Monday deadline, when Gazprom says Kiev must prepay to secure supplies or face a cut-off.
"There are differences of opinion. We are talking about billions. There are major business interests that are at odds with each other," Energy Commissioner Guenther Oettinger said after Wednesday's talks.
"But we have both sides here in Brussels, and that's very encouraging."
If supplies are cut to Kiev, there could be knock-on disruptions to the European Union as roughly half of the gas it receives from Russia - about 30 percent of its gas demand - is shipped via Ukraine.
The talks between Russia, Ukraine and the European Commission have been going on while Kiev, Moscow and the Organization for Security and Cooperation in Europe discuss peace proposals put forward by Ukraine's new president, Petro Poroshenko, to settle the armed stand-off in the east.
Those talks have produced what Kiev says is a mutual understanding on key aspects of the peace plan, intended to end the insurgency by rebels who want parts of Russian-speaking eastern Ukraine to join Russia.
Analysts say resolving the gas row would go some way to taking the heat out of the wider conflict.
Moscow dropped Ukraine's gas price from $485 per 1,000 cubic metres to $268.50 after then-President Viktor Yanukovich turned his back on a trade and association agreement with the EU last year, but reinstated the original price after he was ousted.
Moscow says Ukraine has racked up more than $4 billion in debts to Gazprom and must pay off some of that before a new price deal can be reached.
Kiev paid off $786 million at the start of last week but has since said it will only pay more when an overall deal is struck. (Additional reporting by Natalia Zinets, Pavel Polityuk and Timothy Heritage in Kiev, and Elizabeth Piper and Vladimir Soldatkin in Moscow, Editing by Tom Heneghan and Will Waterman)