The Greek prime minister considers Russia a "strategic partner," and hopes his three-day visit to Moscow will cement ties between the two nations.
Prime Minister Costas Karamanlis congratulated Putin on the result of the parliamentary election held earlier this month, in which Putin's United Russia party won 70 percent of the seats in the legislature.
"The historic ties between our countries are getting stronger, especially in the sphere of energy, which is particularly important for us," Karamanlis told reporters.
Putin said Athens was interested in doubling its natural gas imports from Russia through the proposed South Stream pipeline between 2016 and 2040.
Karamanlis in June announced his country's involvement in South Stream, which would bring Russian gas across the Black Sea to Bulgaria and on to other European countries. The pipeline is to come on stream in 2013.
After meeting with Putin, the Greek leader was set to hold talks with Russia's minister of industry and energy, and the chief executives of Gazprom, the state-owned natural gas monopoly, and Russia's oil pipeline monopoly, Transneft, Interfax reported.
The Greek leader's three-day visit is set to include negotiations for Greece to buy more than 400 Russian tanks. The talks take place amid fresh strains between Moscow and NATO countries after Russia suspended participation this week in the Conventional Forces in Europe arms control treaty.
Greece remains keen to keep pace militarily with Turkey, its regional rival. It has added Russian weapons to its arsenal, including the S-300 surface-to-air missile system.
On Friday, Karamanlis rejected suggestions he was trying to appease Moscow with the tank deal. Greece recently supported a natural gas pipeline project that would bypass Russian suppliers, transporting natural gas from Azerbaijan to Italy.
Also on Tuesday, Transneft said in a press release that Russia, Bulgaria and Greece signed an agreement create a company to build the Burgas-Alexandroupolis oil pipeline, Interfax reported.
The pipeline would create an alternative route for transporting Russian oil, and bypass the congested Bosporus. Under the agreement, Russia will have a 51-percent stake in the company.