"I think we had good impressions on each other. Our talks will continue," former Russian football chief Mutko, who is also a member of FIFA's executive board, told local media.
"Because of a limited time frame I think we will name a new coach early next week, give or take a few days to complete all the paperwork."
The 66-year-old Italian, who quit as England boss in February, is seen by many as the front-runner for the job, which became vacant after Dutchman Dick Advocaat's departure following Russia's failure to get past the group phase at Euro 2012.
Most local outlets reported on Friday that despite having several other high-profile candidates on its list, the Russian FA (RFU) was all but decided on Capello.
"I think Capello is genuinely interested in working in Russia," said acting RFU chief Nikita Simonyan.
"He also is willing to live in Russia full time, not just come here for a few days each month. It says a lot about his intentions to work here."