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.