The City manager's future has been the subject of speculation in recent months and the Russian side are planning a swoop to make the Italian their new coach.
Despite spending over £60m last summer on players such as Hulk and Axel Witsel, Zenit are currently third in their domestic league and club officials want to replace current boss Luciano Spalletti, after a series of disappointing results.
Mancini signed a new five-year deal with City last summer.