Ter Stegen has also been a reported target of Liverpool, and Pep Guardiola is said to want the German at the Etihad Stadium.
The German has been linked with a move away from the club as he is only usually selected for Champions League and Copa Del Rey fixtures, with Barca boss Luis Enrique preferring Claudio Bravo in La Liga encounters.
Liverpool have been among the sides linked with a move for the stopper, but the report says that City are willing to pay a bigger fee than any of their Premier League rivals in pursuit of ter Stegen.
It also states that incoming manager Pep Guardiola wants better cover for Joe Hart than Willy Caballero, despite the heroics of the Argentine in City's League Cup final win on penalties over Liverpool.
There seems to be a fair bit of interest in ter Stegen, but one would question whether he would leave Barcelona for Manchester City just to sit on the bench behind Joe Hart given his situation at his current club. If that is the case, it could be a test of the type of lure that Guardiola will have.
It's inevitable that players will be attracted to work with the Spaniard, but whether it's a case of not playing regularly remains to be seen.
It will also be interesting to see if this springs any of the other clubs interested in the goalkeeper into action, such as Liverpool, as to whether they step up their interest and actively pursue the German.