The former England manager has not received any money since June and said earlier this month he was "getting close to the limit" of his patience.
The Italian is the highest-paid international coach, earning almost twice that of England's Roy Hodgson.
"We have no money to pay Capello's salary," Sergei Stepashin, a member of the executive committee, admitted.
"It's wrong, but when they signed the contract they should have thought through how it's going to be financed. Today we need to look for sources of funding."
Capello signed a contract extension last January, which ties him to Russia until they host the World Cup in 2018.
Since then, though, the Russians failed to win a game at the World Cup finals in Brazil - drawing 1-1 with both Algeria and South Korea, as well as losing 1-0 to Belgium - and were knocked out after finishing third in their group.
Their Euro 2016 qualification campaign started with a 4-0 win over Liechtenstein but has stuttered since, with draws against Sweden and Moldova.
They face group leaders Austria on Saturday, but are preparing for the game without two members of Capello's coaching staff after fellow Italians Christian Panucci and Massimo Neri refused to travel to Vienna.
Speaking to Moscow-based news agency Interfax, former Prime Minister Stepashin echoed the suggestion of the right-wing Liberal Democrat party that Capello and his staff should receive performance-related pay.
"As the former chairman of [Russia's] Accounts Chamber, I believe you should pay for results," said the politician, who served under president Boris Yeltsin. "This would be a more effective use of funds."
The Russian Football Union (RFS) has admitted that it cannot afford to pay coach Fabio Capello's £7m salary.