MOSCOW — When the European Union said it would bail out Cypriot banks by seizing a percentage of deposits, Cypriots erupted. Russian government officials also raged, on behalf of Cyprus’s many Russian depositors.Meanwhile Gazprom, the giant Russian energy company, quietly acted by offering a private bailout plan. Rather than tax deposits, Cyprus could raise money to right its economy by selling Gazprom exploration rights to offshore gas deposits in the Mediterranean Sea.

The fate of this proposal is uncertain. Gazprom refused to confirm it even made an offer. But it illustrates how a sprawling, wealthy company so deeply entwined with President Vladimir V. Putin of Russia that it is often called a state within a state is willing to seize an opportunity and exploit weaknesses and divisions within Europe to cement its position and power.

Gazprom already has vast gas deposits in Siberia. But the emergence of an independent gas industry in Cyprus could further undercut Gazprom’s monopoly pricing power in Europe, already threatened by the global gas glut from the American shale gas boom.

Ownership of Cyprus’s promising though undeveloped reserves, lying beside similarly large deposits found recently off the coast of Israel, would prevent potential competitors from obtaining them and ensure a supply of gas — and Gazprom’s continued power — for generations to come.

Gazprom, the world’s largest natural gas company, accounts for about a tenth of Russia’s gross domestic product as it earns billions of rubles by providing Europe with about 40 percent of its imported gas.

Often, its resources become the Kremlin’s tool of choice for settling domestic and foreign policy problems, as it did in 2004 in a dispute over gas prices and transshipping gas to Western Europe. After a pro-Western government came to power in Ukraine in 2004, Gazprom twice shut off the supply of natural gas to the country at the peak of the heating season. Some countries farther west along the pipelines also ran low on heating fuel, in a sign of the reach of Russian pipeline politics.

While the Gazprom proposal was widely interpreted as an effort to elbow aside the European Union and the International Monetary Fund, it was at the very least audacious: a private company was in effect offering to save a nation’s economy.

On Sunday evening, a day after the European Union announced its plan, the banking subsidiary of Gazprom, called Gazprombank, owned by the employee pension fund, had, according to a Russian news agency, delivered its proposal to the office of the president of Cyprus.

Gazprombank’s maneuver, while clearly aimed at benefiting the parent company, also highlighted the deep dependence of Russia’s business and political elite on Cypriot offshore banking. They have used it to avoid taxes and political risk at home and to access Cyprus’s relatively reliable court system to adjudicate disputes. Russian depositors in Cypriot banks risk losing about $3.1 billion to what the European Union is calling a stabilization tax on bank savings, from a total of $31 billion held by Russians in Cypriot banks, according to a report by Moody’s, the rating agency. Mr. Putin called the tax on deposits “unfair, unprofessional and dangerous.”

Dimitry Afanasiev, the chairman of Egorov Puginsky Afanasiev & Partners, a law firm that advises Russian companies on Cypriot investments, said in an interview, “My understanding is that Gazprom has suggested a private bailout” of Cyprus’s banking system.

A Cypriot television station, Sigma TV, reported that after Gazprom delivered its offer to the office of President Nicos Anastasiades of Cyprus on Sunday evening, Mr. Anastasiades did not hold talks on the offer.

The Cypriot Parliament on Tuesday overwhelmingly rejected the 10 billion euro bailout package that would have placed a tax on bank deposits. It was unclear, however, whether a Russian alternative might still be considered viable.

Gazprom’s spokesman, Sergey Kupriyanov, denied the gas company made the offer. Separately, however, an unidentified company spokesman clarified to the Itar-Tass news agency that a banking subsidiary, Gazprombank, was indeed in talks with the Cypriot government.

The fate of the proposal is as murky as the proposal itself. The Cyprus government wanted American energy companies to develop its offshore assets as a hedge against possible Turkish meddling, said Mr. Afanasiev, who has been a vocal advocate for a Russian alternative to the European proposal.Gazprom and Gazprombank are commercially entwined but under distinct managements; the bank, one of Russia’s largest, has since 2007 been managed by a tight coterie of businessmen with longstanding ties to Mr. Putin.

Gazprom in 2007 transferred control of 47 percent of Gazprombank to its pension fund, Gazfond. The pension fund in turn hired an asset management agency belonging to another bank, Rossiya Bank, to run Gazprombank. This skein of financial transactions, Russian style, resulted in Rossiya Bank, based in St. Petersburg, controlling Gazprom’s financial arm, which over the weekend made the offer to Cyprus’s president.

A former neighbor of Mr. Putin’s at a summer home community outside St. Petersburg, Yuri Kovalchuk, co-founded Rossiya Bank in the 1990s. Another member of that summer home community, called the Ozero co-operative, Nikolai Shamalov, is a major shareholder in the bank.

The press office of Rossiya Bank did not respond to questions submitted in writing on Monday.

Though not widely publicized, the Russian proposal to prop up Cyprus with assets belonging to the Gazprom pension fund was apparently taken seriously enough by Germany’s chancellor, Angela Merkel. Her office issued a statement on Tuesday noting she had warned the president of Cyprus in a telephone call not to consider alternatives to the European bailout; Russia’s offer is the only known alternative.

Michael Olympios, chairman of the Cypriot Investors Association, said one possibility under active consideration was for a Russian bank to buy Cyprus’s biggest troubled lender, the Cyprus Popular Bank, in a deal that could reduce the amount of the 10 billion euro bailout sought by Cyprus. Any such move would most likely be backed by the Kremlin, Mr. Olympios added, and could reduce the tax that Russian depositors might otherwise have to pay.

Russian officials were preparing for talks in Moscow on Wednesday with the Cypriot finance minister, Michalis Sarris, who was expected to request that Russia postpone the maturity date on a 2.5 billion euro loan that it extended to Cyprus in 2011.