The former England manager’s position has come under serious scrutiny following Sunday’s 1-0 home defeat to Austria in a European Qualifier.
Russia are now third in their Euro 2016 group having won just two of their last 10 competitive matches – one of those against Liechtenstein and the other awarded by default when a match against Montenegro was abandoned due to crowd trouble.
Capello, 68, has been in charge of Russia since 2012 and received a four-year contract extension last year to lead them through their home World Cup in 2018.
However, frustrated fans have now set up a fundraising website called ‘Fabio Go Home’ and are trying to oust the manager.
A statement on the site said: “Recent public surveys show the vast majority of the country’s football lovers want Capello to step down.
“But to do that Russia’s Football Union (RFU) needs to pay Capello a serious penalty for the cancellation of his contract – 21.4m euros.”
Russia failed to win a match at the World Cup last summer and are looking likely to be, at best, in a play-off for a place at Euro 2016.