Off contract in June and refusing to consider new terms offered by PSG, Rabiot is free to discuss and sign a pre-contract with a foreign club from January 1.
Barca have been in contact with Rabiot since the end of last season, where they see the France international as an alternative to Ajax midfielder Frenkie de Jong.
However, Sport says the Catalans have become alarmed after learning City are also speaking with Rabiot's camp. Indeed, it's suggested a City representative recently held face-to-face talks with the player.
Giving an edge to City is Rabiot spent time with the club as a youth teamer and is also a fan of manager Pep Guardiola.