"Gref is being viewed as the only candidate," he said, adding that no official nomination had been made yet.
He said that before a shareholders' meeting elects the new head of Sberbank, the Supervisory Board will appoint an acting CEO, indicating that the role could be entrusted to Alla Alyoshkina, deputy head of the board.
On Monday, Russian President Vladimir Putin accepted Prime Minister Viktor Zubkov's proposal to appoint Sberbank CEO Andrei Kazmin as head of the country's postal service.
In response to Zubkov's request to place Kazmin in charge of the service, which is desperately in need of modernization, Putin said: "I have no objections."
Kazmin has held the top position at Russia's largest state-owned bank since 1996.
The Federal Postal Service has around 415,000 employees and 40,000 offices, double the number of Sberbank offices.
Russian Post annually handles over 1.4 billion letters, 38 million parcels and more than 188 million money transfers. The service is currently headed by Igor Syrtsov.