Both countries have sworn that a failure to reach a deal would not affect transit flows of Russian natural gas to Europe, which have twice been cut in the past decade when Moscow and Kyiv haggled over the terms of gas deliveries to Ukraine.
"We need to seek a compromise, a mutually beneficial decision," Azarov said after the talks at Putin's Novo Ogaryovo residence outside Moscow.
"I hope that today's talks will give a decisive impulse in the search for a final resolution."
High-level meetings with Putin are often a signal that a deal is in the works, but the late-night press event was a brief, low-key affair. Putin said it was important to "propose a path for development" of the sector.
Gas crisis possible?
Government and industry sources had played down the prospects for a deal, which means Europe's energy industry, which lost some or all of its Russian supplies in 2006 and 2009, may be on edge until the New Year and beyond.
There is no clear deadline for a deal, which is seen as important to securing shipments of gas to Europe by establishing a compromise price and joint control of the transit pipelines that carry more than half of Russian gas deliveries to Europe.
But Russian gas export monopoly Gazprom has little incentive to hurry with a deal while it stands to gain leverage in the talks as Ukraine's fiscal position worsens. It also continues to collect the higher price stipulated in the current contract in the meantime.
"These talks will go on for a very long time," said one Moscow-based industry analyst and long-time Gazprom watcher. "Even when they announce that they've 'agreed', they'll have to talk for another year."
Azarov, whose government is struggling with a huge trade deficit and has borrowed money from a Russian gas industry bank to pay its November gas bill, told reporters in Prague on Monday a deal was just days away.
"There was lots of movement a week ago, but now it's all quiet," a source in Russia's Energy Ministry announced earlier.
Azarov's government, however, went ahead last week with a 2012 budget that did not include a price cut.
"From pure logic and economics, Ukraine can't sustain this price any more," said a financial source who does business with both governments.
"So in my opinion some kind of arrangement must be reached, otherwise Ukraine will be in real trouble in 2012."