The ex-Soviet Union and Russia keeper, 52, coached Legia Warsaw to the Polish league and cup double last term before leaving by mutual consent.
Leonid Slutsky resigned as Russia coach in June after two defeats at Euro 2016 saw them go out at the group stage.
Russia were named as 2018 World Cup hosts by Fifa in 2010.
Stanislav Cherchesov has been named Russia's new head coach and given a target of leading the hosts to the 2018 World Cup semi-finals.