Several Russian media outlets said Capello, who quit as England boss in February, would arrive on Thursday to negotiate with the Russian Football Union about the vacant position.
Dutchman Dick Advocaat's contract expired following Russia's failure to reach the quarter-finals at Euro 2012 last month.
Russian Sports Minister Vitaly Mutko, who is also a member of FIFA's executive board, said on Wednesday the RFU would name a new national team coach by July 20.
The RFU also needs a new president after Sergei Fursenko resigned last week following Russia's disappointing results in Poland and Ukraine.
Capello visited Moscow last November when rumours linked the 66-year-old Italian with the coaching role at wealthy Russian club Anzhi Makhachkala.
At the time, Capello said: "I came to Moscow to see the ballet at the Bolshoi Theatre, not for football."