The submarine, the third India has leased from Russia, would be delivered by 2025, AFP reported on Friday.
The deal has yet to be officially announced by the governments in India and Russia.
Russia remains a major supplier of arms to India, irking the United States, which has imposed sanctions on nations buying military hardware from Moscow.
The US had warned India over buying Russian arms which it said would be in violation of the sanctions against buying weapons from Moscow
Last October Russian President Vladimir Putin and Indian Prime Minister Narendra Modi met and inked a deal for Delhi to buy Russia's S-400 surface-to-air missile defense system for $5.2 billion.