The destroyer is part of a Northern Fleet task force, currently on a tour-of-duty in the Mediterranean, led by the Admiral Kuznetsov aircraft carrier.
However, officials from the Northern Fleet have so far refused to confirm or deny the purpose of the ship's ultimate mission.
"We will not announce the goals and tasks set for the Admiral Levchenko in the near future," said Capt. 1st Rank Igor Babenko, a fleet spokesman.
"We can only confirm that Admiral Levchenko destroyer has passed through the Suez Canal and entered the Indian Ocean," he added.
The destroyer will join up with the Pyotr Veliky missile cruiser and a task force from Russia's Pacific Fleet, comprising the Admiral Vinogradov, an Udaloy class destroyer, a tugboat and two fuel tankers, which will also take part in the INDRA-2009 naval exercise with the Indian navy.
INDRA is a biennial Russian-Indian exercise aimed at practicing cooperation in enforcing maritime law and countering piracy, terrorism and drug smuggling. INDRA-2009 is the fourth such exercise since 2003 and will involve a number of live-fire drills.