The crew will set sail for India toward the end of January in the 8,000-ton submarine that will be renamed Chakra, The Times of India said.
Nuclear submarines are operated also by the United States, Russia, Britain, France and China. India briefly operated a nuclear submarine form 1988-91 when it leased a Charlie class vessel from Russia. It, too, was called the Chakra while in service with the Indian navy.
The Akula II class Nerpa submarine will be with the Indian navy under a 10-year lease from the Russians, The Times of India report said.
Handover of the Nerpa was completed during a signing ceremony at the Bolshoi Kamen shipbuilding yard in Bolshoy Kamen, a small coastal town whose main activity is construction and repair of nuclear subs.
Bolshoy is around 25 miles from Vladivostok in the isolated Primorsky Krai region and is off-limits to foreigners.
The Nerpa leaves Russia after a controversial history, including a fatal on-board fire in November 2008 just after the vessel was made operational by the Russian navy.
The vessel was sailing through the Sea of Japan when the firefighting system was accidentally activated, Russian media reported at the time. Three naval personnel and 17 civilian specialists died after inhaling chemical gases, Russia's Defense Ministry said.
Amur Shipbuilding began construction of the Nerpa at their plant in Komsomolsk-on-Amur around 1993. The ship was intended for the Russian navy but delays and financial problems lengthened the construction schedule.
A $650 million deal with Indian government in 2004 rescued the program which by then had been stopped. The Nerpa finally was launched in October 2008 and entered service with the Russian navy in late 2009, pending handover to India.
The Chakra will be joined by another nuclear submarine, the Arihant, built by the Ship Building Center in Visakhapatnam. A launch ceremony for the 6,000-ton, 365-foot Arihant was in July 2009 and the ship with a crew of up to 100 is expected to be operational by the end of this year.
A report by the defense news Web site Defense Professionals in July 2009 said the Arihant design possibly is based on the older 5,000-ton Russian Charlie-II class submarine. Russia decommissioned it last Charlie class sub in 1994.