The deal -- which according to the reports took months to negotiate -- comes as tensions run high between India and Pakistan following their biggest standoff in years, and as Chinese influence grows in the region.
A defence ministry spokesman declined to confirm the agreement to AFP but the reports said that the submarine, the third India has leased from Russia, would be delivered by 2025.
Russia, India's Cold War ally, remains a major supplier of arms to India, irking the United States which has imposed sanctions on nations buying military hardware 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 defence system for $5.2 billion.
But India also shares U.S. fears about China's growing assertiveness in the Indian Ocean, where New Delhi has traditionally held sway.
In 2017 India and China had a military standoff over a Himalayan plateau claimed by both Beijing and Bhutan, a close ally of India.
China has made inroads in Sri Lanka and Maldives, countries that India considers to be in its sphere of influence, through its Belt and Road Initiative (BRI).
India has raised concerns about the initiative as a major section passes through Pakistan-administered Kashmir, a disputed territory which was again the fuse for the latest crisis.
On February 26 India said its warplanes hit a militant training camp in Pakistan in response to a suicide bombing in Kashmir, which killed 40 paramilitary officers and was claimed by a militant group based in Pakistan.
Pakistan a day later carried out its own air raid, triggering a dogfight in which an Indian plane was shot down. India also says it shot down a Pakistan plane but Islamabad denied this.
Tensions cooled after Pakistan last Friday returned the Indian pilot of the downed aircraft, although both nations have continued to fire shells and mortars over their de-facto Kashmir border.
Russia remains a major supplier of arms to India and in October leaders Modi and Putin met to sign a deal over Russia's missile defence system for $5.2 billion