The 28-year-old Serbian international has been one of the Blues' most consistent performers this season, and has attracted the attention of a number of clubs.
Ivanovic first moved to Stamford Bridge in 2008 and has enjoyed his time in west London, despite a succession of managers coming and going during the past three and a half years.
The Russians are willing to offer Chelsea £8million for the versatile full-back, who can also operate in central defence when needed.
After impressive performances in the Champions League in both legs against Napoli in the previous round, the Serie A side are also believed to now be interested in his signature.
Chelsea owner Roman Abramovich is expected to continue the planned overhaul of the club's playing squad this summer, but Ivanovic is not one of those senior stars poised for the exit door.