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Widespread reports in Moscow claim the defender-cum-midfielder is on the brink of securing a £13million move to Anzhi, whose defender, Rasim Tagirbekov, has claimed Zhirkov has met with the club's squad and head coach Gadzhi Gadzhiev.
Tagirbekov said: "Of course, Yury is a great player. Everybody in the team is very glad that he will join our club.
"This transfer will strengthen Anzhi a lot. Zhirkov has already managed to talk to the team's head coach Gadzhi Gadzhiev and practically all the players."
According to Bobsoccer.ru, Zhirkov himself confirmed he had travelled to Russia, saying: "I have arrived for two days to spend my weekend here."
The sale of Zhirkov would free up space in Chelsea's 25-man Premier League squad for another foreigner, with the club weighing up a third bid for Tottenham's Luka Modric.
Zhirkov, who turns 28 this month, has struggled to establish himself in the Blues' first team since moving from CSKA Moscow two years ago.
A knee injury hampered his progress but he has ultimately failed to depose Ashley Cole as first-choice left-back or force his way into midfield.