The news agency says the Russian parliament plans to vote in June on a draft law giving the company exclusive rights to export gas.
Reuters says it has obtained a copy of the draft, which says gas should be considered a strategic material and only exported by state-controlled Gazprom, to protect the national interest.
The draft also says that having one sole "efficient" operator would avoid competition between Russian gas suppliers in foreign markets, which could lead to a decline in prices.
The European Union has repeatedly called on Russia to open its gas market to boost competition and potentially bring down prices. It also says liberalizing Russia's gas market would prevent disruptions in supplies.