Following the team's 1-0 loss to Austria in a Euro 2016 qualifier on Sunday, the 68-year-old Italian's days in the job appear to be numbered.
Capello's side are third in the Group G standings, eight points behind leaders Austria.
"We have already made a certain decision on Capello's future," Russian sports minister Vitaly Mutko told the TASS news agency.
"But there is the executive committee of the Russian Football Union (RFS). I think in the very near future one of these bodies will look at this question in detail," he added.
Mutko plans to attend the RFS executive committee meeting on June 24 at which Capello's future will be discussed.
Capello refused to answer questions about his job following the defeat by Austria.
"I don't have a problem with talking to my employers about this question," he told a news conference.
"However, I have worked for eight months for free. Now they owe me for another two months. This is not a problem for me. If I was only interested in the money then I would have complained to FIFA a long time ago. I have not done this once out of respect to Russia and the RFS."
Since the start of last year's World Cup in Brazil, Russia have won just two of 10 competitive internationals under the Italian. They beat Liechtenstein 4-0 and were awarded victory over Montenegro after the match in Podgorica was abandoned due to crowd trouble.
Capello is under contract with Russia until the end of the 2018 World Cup and media reports said it would cost the RFS 21.6 million euros ($24.26 million) in compensation to sack him.
The RFS is 1.4 billion roubles ($26.09 million) in debt, according to media reports.
(Editing by Ed Osmond)
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