Russian Football Union acting president Nikita Simonyan told the R-Sport agency Capello's role was in question.
Simonyan's comments came following Russia's 1-0 defeat by Austria in a Euro 2016 qualifier on Sunday.
Russia are third in their group and are facing the prospect of needing to win a play-off to reach the tournament.
Former England manager Capello, 68, took over as Russia boss in July 2012 and signed a four-year contract extension in January 2014.
The deal was designed to keep the Italian in the role until the 2018 World Cup, which is due to be staged in Russia.
Russia were knocked out at the group stage of the 2014 World Cup in Brazil without winning a game, and have eight points from six games in their Euro 2016 qualifying campaign.
Capello's side have failed to win in five games, losing two, and are eight points behind Group G leaders Austria and four adrift of Sweden, in second.
Fabio Capello's future as Russia manager is in doubt with the nation's football chief due to "discuss" whether to sack him.