Kremlin spokesman Dmitry Peskov told news agencies that President Vladimir Putin had been informed of the situation and was being kept updated on the search operation.
Local news agencies, citing the defence ministry, said the Tu-154 plane had crashed shortly after take-off at 5:40 am local time (0240 GMT) from the southern city of Adler where it had been refuelling.
Defence ministry spokesman Igor Konashenkov told Russian news agencies that one body had been recovered six kilometres off the coast of the resort city of Sochi, as a frantic search operation continued to hunt for the missing.
"Fragments of the Tu-154 plane of the Russian defence ministry were found 1.5 kilometres from the Black Sea coast of the city of Sochi at a depth of 50 to 70 metres," the ministry said.
Among the plane's 84 passengers were Russian servicemen as well as 64 members of the Alexandrov Ensemble, the army's official musical group also known as the Red Army Choir, and its conductor Valery Khalilov. They were headed to Syria to participate in New Year celebrations at the airbase.
There were also eight crew members onboard, the ministry said.
Nine journalists were among the passengers, with state-run channels Pervy Kanal, NTV and Zvezda saying they each had three staff onboard the flight.
A list of passengers published by the defence ministry also included Elizaveta Glinka, a doctor and charity worker who serves on the Kremlin human rights council.
Mikhail Fedotov, who heads the council, said Glinka was travelling to Syria to bring medication to a university hospital in the coastal city of Latakia near the airbase, agencies reported.
"It's too early to say anything," agencies quoted Peskov as saying, adding that Putin was in constant contact with Defence Minister Sergei Shoigu.
"The president is waiting for the picture to be clear."
Konashenkov said that Deputy Defence Minister Pavel Popov had flown to Adler along with a team tasked with clarifying the circumstances surrounding the crash.
Russia's Investigative Committee said a criminal probe had been launched to determine whether violations of air transportation safety had led to the crash.
Investigators are currently questioning the technical personnel responsible for preparing the plane for take-off, the committee said.
Tu-154 aircraft have been involved in a number of accidents in the past.
In April 2010 many high-ranking Polish officials, including then president Lech Kaczynski, were killed when a Tu-154 airliner went down in thick fog while approaching the Smolensk airport in western Russia.
In October, Putin approved a law ratifying Moscow's deal with Damascus to deploy its forces in the country indefinitely, firming up Russia's long-term presence in Syria.
Russian warplanes have flown out of the Hmeimim base to conduct air strikes, and the base is also home to an S-400 air defence system.