The Brazilian international was signed by City last summer in a £3.3m transfer from Inter Milan.
But a series of knee and ankle injuries have interrupted his first season at the Etihad Stadium, and the two big-spending Russian clubs are thought to be readying summer bids for the full-back.
The 31-year-old has failed to dislodge first-choice right-back Pablo Zabaleta when fit, and he's made just four league starts this term.
The report claims Roberto Mancini could be willing to let him go at the end of the season, and according to Russian newspaper Izvestia, Anzhi and Zenit would both be interested in signing him permanently.
City would want at least double the fee they paid Inter Milan for Maicon, and a £7m asking price is thought to be on the table.
Maicon is under contract with the defending Premier League champions until the end of next season, but with Micah Richards returning from long-term injury, Roberto Mancini is unlikely to need three right-backs.
That could allow the Brazilian to move on if the price is right.
Maicon was once regarded as the best right-back in the world, and was superb for Inter during Jose Mourinho's reign at the San Siro, helping the Italian club to a Champions League title.
But he hasn't been able to replicate that form since moving to City.