Reports in both Italy and Russia claim that Capello has arrived in Moscow to discuss the Anzhi post just days after quitting as England boss.
Capello resigned his role with the Three Lions on Wednesday following his criticism of the Football Association's decision to strip John Terry of the England captaincy.
The 65-year-old was a target for mega-rich Anzhi during his time in charge of England and he could now take over the ambitious Russian club.
According to reports, Anzhi have sacked current boss Yury Krasnozhan, although there has been no official confirmation of the decision.
Krasnozhan only became Anzhi boss in December and the club say that his future will be decided on Monday amid the growing speculation concerning Capello.
"There will be changes in the club staff," said Anzhi deputy president German Chistyakov. "We have serious reasons for those changes."
Speculation also suggests that Anzhi officials have travelled to Italy to hold talks with Inter Milan regarding a stunning swoop for Dutch playmaker Wesley Sneijder.
Anzhi have previously signed Samuel Eto'o from the Serie A club and the Russian transfer window remains open until later this month.