Russian Sports Minister Vitaly Mutko revealed discussions with the former England boss were ongoing ahead of a possible appointment next week.
"I think we had good impressions on each other. Our talks will continue," Mutko reportedly told local media.
"Because of a limited timeframe I think we will name a new coach early next week, give or take a few days to complete all the paperwork."
Capello was named in an ambitious 13-man wish-list earlier this week that also included Harry Redknapp, Pep Guardiola and Rafael Benitez.
The 66-year-old Italian, who had previously been linked with big-spending Russian club Anzhi Makhachkala, has been out of work since he quit as England boss in February.
Russia have been without a manager after Dick Advocaat left the post, to join PSV Eindhoven, after failing to steer his side out of the group at Euro 2012.