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.

Judge Angela Finnerty said the woman should not have brought the child to England until a family court in Japan had resolved a dispute she was having with the her estranged husband.
The judge's decision has been revealed in a written ruling.
It follows a hearing in the Family Division of the High Court in London.
No-one involved was identified.
The father had asked Judge Finnerty to order his son's return to Japan under the terms of international civil legislation relating to child abduction.
Judge Finnerty said the man and woman - who had worked at different universities in Japan - split in 2013 after marrying nine years ago.
They had reached an agreement about sharing care of their son after mediation in Japan.
The woman had then been offered a post at Cambridge and said she wanted to take the child with her. The man objected.
She had flown to the UK with the boy before a court in Tokyo had ruled on the dispute.
"I am satisfied that (the boy) has been retained in the United Kingdom unlawfully in breach of the father's custody rights," said Judge Finnerty.
Fabio Capello's future as Russia manager is in doubt with the nation's football chief due to "discuss" whether to sack him.
[NEXT_CONCEPT]
A Cambridge University academic has been ordered to return her seven-year-old son to his father in Japan, a family court has ruled.