The Nerpa, under the command of an Indian crew, left its base on Russia's Pacific coast earlier this week, bound for an Indian naval base, Interfax news agency quoted a source in Russia's military-industrial complex as saying.
Another RIA report, citing the plant that built the Nerpa, said the submarine had not yet been officially handed over to India. It said the Indian crew was only training aboard the Nerpa at sea.
Russia's Defence Ministry and navy could not be immediately reached for comment. The press service of the Pacific Fleet declined from divulging any details.
Building of the Nerpa began in 1993 but the submarine was only launched and started sea trials in 2008 due to the piecemeal funding of its construction.
In Nov 2008 the submarine was on sea trials when its fire extinguishing system switched on unexpectedly. 20 people died after inhaling the toxic gas used as a fire suppressant, authorities said.