The People claim City boss Roberto Mancini is keen on signing the £30million-rated Frenchman as he expects to lose out in the race to sign Franck Ribery.
Gourcuff has been a hit since joining Bordeaux from AC Milan last summer.
The 23-year-old has been hailed as the new Zinedine Zidane in the French press this season.
Manchester United have also reportedly made checks on the player several times this season.
