RFU acting president Nikita Simonyan said they had decided to hand the job to Capello, known as a strict disciplinarian, following Russia's disappointing Euro 2012 campaign.
"We have decided to name Fabio Capello as the new manager of our national team. We await him in Moscow over the next few days for the final details and the signing of the contract," Simonyan told local media on Monday.
The 66-year-old Italian, who quit as England boss in February, said he was happy to work in Russia.
"I am happy, if contract negotiations go well as I believe they will, it will be a wonderful adventure, as Russia is a great country," he told Italian news agency ANSA.
Capello will succeed Dutchman Dick Advocaat, whose contract expired after Russia failed to advance past the group phase At Euro 2012.
The Russians had been hoping to reach at least the quarter-finals in Poland and Ukraine following their surprise run to the Euro 2008 semi-finals under Advocaat's compatriot Guus Hiddink.
Upon returning home, the players faced heavy criticism from high-ranked politicians, media as well as ordinary fans.
"We had a good team but order and discipline was obviously missing," Russian Sports Minister Vitaly Mutko said last week.
Capello, who has had success coaching top clubs such as Real Madrid, AC Milan, AS Roma and Juventus, had already received a glowing endorsement from Russia captain Andrei Arshavin.
"Russia's national coach has been appointed - he is Fabio Capello. We wish him success in his new job," Arshavin, widely considered the country's most influential player, wrote on his personal website.
Capello, who will become Russia's third foreign coach in a row, will be tasked with blooding new players into an ageing team, something his predecessor Advocaat failed to do in his two years at the helm.
Capello was regarded as favourite for the job even before coming to Moscow for talks with Simonyan and Mutko last week after the RFU published a list of 13 high-profile candidates, which also included former Tottenham Hotspur manager Harry Redknapp, ex-Liverpool boss Rafael Benitez and former Barcelona coach Pep Guardiola.
"Capello would be a good choice," Mutko said last week.
Russian media reported Capello would earn up to 10 million euros a year after signing a contract through to the 2014 World Cup, with the possibility of extending it for another two years.